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AN 


ELEMENTARY TREATISE 


ON THE 


THEORY OF DETERMINANTS. 
A TEXT-BOOK FOR COLLEGES. 


BY; 


ay 
PAD beHY HANUS= 
FoRMERLY PROFESSOR oF MATHEMATICS IN THE UNIVERSITY OF CoLORADO; 


Now PRINCIPAL OF DENVER HIGH ScHOOL, 
DIstTRIcT No. 2. 


BOSTON: 
PUBLISHED BY GINN AND COMPANY. 
1898. 


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Nec vena 


caw Naa Vo 


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: yaa Entered, according to the Act of Congress, in the year 1886, by 
ee J PAUL H. HANUS, 
: c ' in the Office of the Librarian of Congress, at Washington. — oe: 


— 


J. 8. Cusnine & Co., PRINTERS, Boston. 


Yriext 


PREFACE. 


THE importance of a knowledge of Derrrerminants to all 
who extend their reading beyond the elements of mathematics, 
and the fact that most modern writers employ the determinant 
notation, have led to the belief that an American work on 
Determinants might satisfy a growing demand. 

This is a text-book, and not an exhaustive treatise. Enough 
is given, however, to enable the student to use the determinant 
notation with ease, and to enable him to pursue his further 
reading in the modern higher mathematics with pleasure and 
profit. 

The book is written with reference to the wants of the 
private student as well as to the needs of the class-room. The 
subject is at first presented with great simplicity. As the stu- 
dent advances, less attention is given to details. More than 
half the volume is devoted to applications and special forms, 
that the reader may get some notion of the power and utility 
of determinants as instruments of research. 

Throughout the work care has been taken to show how each 
new concept has been evolved naturally ; and, whenever it is 
thought advisable, a special case precedes the general dis- 
cussion. 

The work has been written in the far West, where contact 
with others in the same field was practically impossible. I 


1V PREFACE. 


shall therefore be grateful for any notification of errors that 
may have escaped detection. 

My thanks are due to Messrs. J. S. Cusninec & Co., of 
Boston, for great care and patience manifested in the prepara- 
tion of the plates. 

Among the works consulted most assistance has been derived 
from the following. All the works named have been used 


freely. 


Matzka.— Grundziige der systematischen Einfiihrung und Begriin- 
dung der Lehre der Determinanten. 

Baltzer.— Theorie und Anwendung der Determinanten (Fiinfte 
Auflage). 

Giinther.— Lehrbuch der Determinanten-Theorie (Zweite Auflage). 

Diekmann. — Einleitung in die Lehre von den Determinanten und 
ihrer Anwendung auf, etc. 

Dostor.— Eléments de la Theorie des Déterminants avee Applica- 
tions, etc. (Deuxieme edition). 

Hoiiel. — Cours de Calcul Infinitésimal. 

Scott.— A Treatise on the Theory of Determinants and their Appli- 
cations, etc. 

Burnside and Panton.— The Theory of Equations, with an Intro- 
duction, etc. 

Muir. — A Treatise on the Theory of Determinants. 


I am especially indebted to the last two works for many 
examples. | 
PAUL H. HANUS. 


BOULDER, Cou., May, 1886. 


16-19. 
20. 


21. 


99 


at hed 


23-24. 
25. 
raf 
27. 


CONTENTS. 


CHAPTER I. 


PRELIMINARY NOTIONS AND DEFINITIONS. 


PAGE. 
Discovery of Determinants . , : : : ° 1 
Determinants produced by eliminating the unknowns 
from a system of simultaneous equations : : 2-8 
Values of the unknowns in determinant form . . 8-10 
Change of sign. : ‘ ‘ ‘ ; : ; 9 
Notation . : , : ‘ ; : ; . 11-12 
Expansions with square notation : ; : : 13 


Rule for expanding a determinant of the third order . 14 
Examples. ‘ t 3 : ; : : . 14-16 


CHAPTER II. 


GENERAL PROPERTIES OF DETERMINANTS. 


Definition and notation : : : : ‘ . 17-20 
Corollaries . : : , : ‘ ‘ : : 20 
Inversions of order ‘ ‘ , : : . . 20-21 
Number of terms in a determinant . ; , ; 21 
Corollaries ; expansions ‘ : ‘ ; ‘ . 22-23 
If the rows in order are made the columns in order, ete. 23 
Number of positive and negative terms. 5 : 24 
Interchange of two parallel lines . ; ‘ : : 24 


ART. 


59-62. 


CONTENTS. 


Two identical parallel lines . : : : : 

Cyclical permutations . 

Corollary 

Examples 

Every element of a line multiplied by the same 
number 

Corollaries 

Decomposition of determinant with polynomial ele- 
ments 

Converse of 34 

Transformation by addition of parallel lines 

Minor determinants, or Minors 

Expansion of determinant as linear function of the 
elements of one line. 

Coefficient of any element in the expansion of a 
determinant 

Corollaries ; expansions 

Examples ; : ; ; 

Elements of a line multiplied by first minors of corre- 
sponding elements of a parallel line 

Expansion in zero-axial determinants . 

Simplification by taking each consecutive pair of ele- 
ments in the first row, etc. 

Examples 

The product of two determinants 

Examples ; 

Laplace’s Theorem (expansion) . 

Product of a determinant by one of its minors 

Rectangular Arrays or Matrices (product of) 

The Reciprocal or adjugate determinant 

Examples : 

Special expansions (including Cauchy’s Theorem, 
Example III.) 


PAGE. 


25-26 


nr = 


ART. — 
64. 
65-66. 
67-70. 
71-72. 
73. 
74-78. 
(KF 

79. 

80. 

81. 
82-83. 
84-89. 
90-100. 
91. 

92. 

93. 
94-95. 
96. 
97-100. 
101-102. 
103. 


CONTENTS. 


Solutions of certain determinant equations . ° 
Differential of a determinant : ; 3 : 


CHAPTER III. 
APPLICATIONS AND SPECIAL Forms. 


Solution of a system of simple simultaneous 
equations . : : 

If the equations of the system are not indepen- 
dent . 

If m, =m, = ++» =™m,_1=0, and one m as m, does 
not ° : . : é 

Solution of a system of linear homogeneous 
equations . : ; ; : . : ; 

The condition 4=0 for a system of homo- 
geneous equations - 

Condition fulfilled when n equations containing 
n—1 unknowns are simultaneous 

Solution of a system of simple equations by 79 

Another application of 79 

The Matrix as the result of elimination 

Applications of preceding processes 

Resultants, or Eliminants 

Euler’s Method of elimination 

Sylvester’s Method 

Bezout’s Method (Cauchy) . 

Resultant in terms of the roots 

Properties of Resultant ; : : : 

Applications of Sylvester’s Method . : 

Discriminant of an equation 

Resultant of a system of homogeneous equations 
when'n—1 are linear, and one is of the second 


degree 


Vili 
PAGE. 


76-78 
79-81 


82-86 


86 


87 


87-92 


91 


92-93 
93-94 
94 
95-97 
98-110 
110-126 
111-112 
113-114 
114-118 
118-121 
121-122 
122-126 
126-129 


129-136 


Vill 


ART. 


104-105. 
106. 
107-112. 


113. 


314-118. 
119-126. 


127-131. 
132-154. 

135. 
136-150. 
4151-163. 
164-170. 

ait 
172-176. 
177-179. 


CONTENTS. 


Special solutions of simultaneous quadratics 

Solution of the Cubic 

Symmetrical determinants, definitions, and some 
special properties _ . 

If x be subtracted from each element of the prin- 
cipal diagonal of a symmetrical determinant . 

Orthosymmetric determinants : , 

Skew determinants and skew symmetrical deter- 


minants , . ; ; : : : 
Pfaffians 
Circulants . : : : A £ : : 
Centrosymmetric determinants . ° . ‘ 
Continuants . : : ; : : : . 
Alternants . , ; ; : - : ‘ 


The Jacobian , ; : : : ‘ ‘ 
The Hessian . ; ; : p - : , 
The Wronskian . : : : : : : 
Linear substitution; Orthogonal substitution . 


PAGE. 
131-136 
187-188 


138-144 


144-146 
146-150 


151-159 
159-164 
164-169 
169-171 
171-186 
187-201 
201-209 

209 
209-212 
213-217 


THEORY OF DETERMINANTS. 


CHAPTER I. 
PRELIMINARY NOTIONS AND DEFINITIONS. 


1. The first notion of Determinants we owe to Leibnitz, who, 
in his attempts to simplify the expressions arising in the elimi- 
nation of the unknown quantities from a set of linear equations, 
employed symbols nearly identical with our present determinant 
notation. In a letter dated April 28, 1693, Leibnitz communi- 
cates his discovery to L’ Hospital; and later, in another letter, 
expresses the conviction that the functions will develop remark- 
able and very important properties, — a conviction which time 
has abundantly verified. Leibnitz, however, never pursued the 
subject himself, and his discovery.lay dormant till the middle 
of the eighteenth century. 

In 1750 the celebrated geometer, Gabriel Cramer, rediscovered 
determinants while working upon the analysis of curves. Dur- 
ing the course of his investigations, Cramer had to solve sets 
of linear equations, and naturally encountered the same func- 
tions that had attracted the attention of Leibnitz.* To Cramer 
is due the general rule for the solution of m simultaneous linear 
equations (non-homogeneous), containing as many unknown 
quantities. | 

This rule was inferred without proof from the form of the 
values of the unknown quantities obtained in solving sets of 
two and three equations. 


* The particular problem which led to Cramer’s discovery of deter- 
minants appears to have been: To pass a curve of the vth order through 


2 
any ey given points. 


2 THEORY OF DETERMINANTS. 


Since the time of Cramer important advances have been 
made. The names of many celebrated mathematicians appear 
in the list of those who aided the evolution of a theory of deter- 
minants. Prominent among these are Vandermonde and Gauss. 
From Gauss the name ‘‘ determinant” instead of ‘‘ resultant” 
was adopted by Cauchy. Cauchy and Jacobi are perhaps to 
be considered as the greatest among those who first developed 
the subject. The monograph of Jacobi, published in 1841,* 
established the foundation of a treatise on the theory of 
determinants; and his.own writings, as well as the works of 
many eminent mathematicians during the past fifty years, attest 
the wonderful power of determinants as instruments of mathe- 
matical investigation, and the fruitfulness of the functions 
themselves. 


2. The most natural way of approaching the theory of deter- 
minants would be along the line of development. This is 
accordingly our purpose. Owing to peculiar difficulties attend- - 
ing this mode of procedure, we can however only employ this 
method at the outset, and must soon adopt a presentation 
better suited to the further unfolding of the subject, and free 
from the peculiar difficulties alluded to. 


Determinants of the second, third, and fourth order. 


3. Consider the set of four simultaneous linear equations : — 
(1) qe+hy+toaz+dyt= m } 
(2) au +tboy +2 +d,t = my | 
(3) ase+bsy +0,2 + dat = mz : 
(4) age thy +az+dt=m, } 
Here it will be convenient to eliminate the unknown quantities 
in a uniform manner, as follows: in each set of equations to be 
obtained, (2) will be multiplied by the coefficient of the un- 
known in (1) that is to be eliminated, and (1) by the corre- 
sponding coefficient in (2); (3) will be multiplied by the 


. . .. . . . 
* De Formatione et Proprietatibus Determinantium. 


PRELIMINARY NOTIONS AND DEFINITIONS. 3 


coefficient of the unknown under consideration in (2), and (2) 
by the corresponding coeflicient in (3); and so on through the 
set. Having thus made the coefficients of one of the unknowns, 
x, say, the same in all the equations, we will then eliminate x 
by subtracting (1) from (2), (2) from (3), ete. We shall find 
in performing these operations that the coefficients of the un- 
known quantities and the absolute term after each elimination 
are functions of a particular form, and subject to the same law 
of formation, —that these functions are, in fact, DETERMINANTs. 
Eliminating x in set I. as directed, we have 


(1) (@ibe— Gab) y+ (Aie2— A901) 2 + (dy — Aol, t= aym.—agmy 
(2) (dybz oa 302) Y 4. (AgC3 Peis (3C2) v4, 4. (ad es AA) t — Ags aaa AsMo } I] 


(3) (d3b4— agg) Y + (gy — MgC) Z+ (gd, — ays) t= aM, — AyMs 


4. Examining these binomial coefficients, we see that each 
contains one positive and one negative term, and involves four 
. quantities, ViZ., Q,, dg, 01, 023 OF Ay, M3, Co, Cz, etc. It will also 
be noticed that each term never contains more than one .a 
(coefficient of x), or b (coefficient of 7), or ¢ (coefficient of z), 
etc., but that each term does contain all the subscripts that 
occur in the binomial. Finally, the terms in which the sub- 
scripts occur in their natural order are positive, while in the 
negative terms there is an inversion of the natural order in the 
subscripts, 7.e., dsc, is +, but a,c; is —. Such binomials are 
determinants of the second order.* (The order of a determinant 
is determined by the number of factors in each term.) It has 
been agreed to denote them, following Laplace, by writing the 
letters involved in regular succession, affecting each with the 
subscripts in order, and enclosing the whole expression within 
parentheses, thus : (@b.) = @, by — db, } (My C3) = AyC3 — Azle, etc. 

Introducing this notation, set II. becomes 

(1) (abs) y + (a, C2) 2 + (ade) t= (4M) 
(2) (dbs) Y + (GeC3) 2 + (yds) t= (a.m) } III. 
(3) (a3b4) y + (GgCq) % + (3d) t= (ag i04) 


* The general definition of a determinant is given in 17, Chap. II. 


4 THEORY OF DETERMINANTS. 


5. If we now eliminate y, according to the directions givep 
in 3, we have 
(1) [(a@,b2) (Gees) — (4203) (Me) 2 + [ (4152) (ads) 
— (dzb;) (dy) ]t = (ay b2) (deitg) — (23) (A M2) 
(2) [(aobs) (gs) — (p04) (es) ]% + [ (420s) (4544) 
— (dgb4) (dads) ]t = (Agbg) (Az 7m4) — (3b4) (42s) 
Examining the binomial coefficients of the unknowns, and 
the absolute terms in set IV, we see at once that they are of 
the same form; and if we can simplify any one of them and 
discover the law of formation, we have them all. For this 
purpose let us expand the coefficient of z, putting, for short- 
ness, this coefficient equal to C. Then, by the definition in 4, 


C = (a,b2) (23) — (obs) (ee) 
= (102) (A2C3 — A3Cy) — (Ayd3) (A, C2 — Mey) 
= ly [ (02) ¢3 + (debs) C1] — Co [ (G1 Bg) Mg + CAD) ay |. 
The last binomial, 

(db2) ds + (yb3) Ay = (by — Ayb,) Ag + (A2b3;—Agb2) Oy 
= Uy (A, 03 — Agb,) = Ay (4,03). 

°. O = dy [(a,02) 3 — (a,b) Co + (2b3) G | 
= (ly [ ty b9C3 — Ay Dy C3 — Ay D3 Cy + Mg, Cy + Cg b3C, — Az 02¢]. 


Here the quantity within brackets consists of 2-3=6 terms, 
i.e., of as many terms as there are permutations of the sub- 
scripts 1,2,3. Three of the terms are positive and as many 
are negative. The quantity involves the 3°=9 quantities a, 
Gn, M3, Dy, Bo, Ds, Cy Coy Coe 

No term involves more than one a, or 0b, or c, but does con- 
tain all of the subscripts 1,2,3, each term containing a different 
permutation of these numbers. Finally, as before, we notice 
that those terms in which the subscripts occur in their natural 
order, or in which there is an even number of inversions* of 


* In a series of integers which are all different there is said to be an 
inversion of order when a greater number precedes a less. Thus in 13452 
there are three inversions, in 21354 there are two inversions, etc. 


PRELIMINARY NOTIONS AND DEFINITIONS. 3) 


order, are positive, while those terms are negative in which the 
number of inversions of order of the subscripts is odd. Such 
a function is a determinant of the third order. A determinant 
in which the quantities are those of C is denoted by (ay Dg Cs). 
We therefore have C = ay(a,b:c;). It must be carefully noticed 
that the equation 


(@, bog) = (4,02) C3 — (1 B3) Cp + (Agds) C 
= A D203 — AyD C3 — Ay Dg Cy + Az Dy Co + Aly dg Cy — Ag d2Cy 

gives the expansion of a determinant of the third order. 

Employing the notation just explained, the coefficient of 
t in (1) is evidently a,(a,b.d;), and the absolute term is 
,(d,b0,m;3). The coefficients and the absolute term of (2) will 
obviously be dz (d2b3¢4) , Az (d,03d4) 5 dg (@2b3™4), in order. 

Introducing this notation into set IV, and dividing (1) and 
(2) by a, and a; respectively, we have 


(1) (a, b9¢3) 2 + (a, b.d5) t = (a, b,ms3) 
(2) (Ggb3c,) 2 + (dy b3d,) t = (d2b,m4) 


6.* If we now eliminate z in the same manner as heretofore, 
we have 


[ (a b2¢3) (2b3d4) — (2b3¢,) (ab. d5)] ¢ ty 
= (dy by €3) (dy b3 74) a (dy bg C4) (ay b,Mz) 


The preceding results naturally imply a simplification and law 
of formation to be discovered in the coefficient of ¢ and the 
absolute term of VI. 

To simplify the coefficient of ¢, which for shortness we will 
call C, as before, we proceed as follows : 


C = (a,)2¢3) (dy b3d,) — (A3¢4) (0,02 ds5) 
= (0 b9C3) [ (gb3) dy — (A204) ds + (A304) de] 
— (igb3¢,) [ (a, b2) ds — (a3) dy + (dbz) dy] 
= (d2d3) [ (0, 09C3) dy — ( Agb3¢4) |] — Dds + D,d, ; 


* 6 may be omitted on first reading, if thought best. 


6 THEORY OF DETERMINANTS. | 


in which 
D = (Gybo¢3) (264) + (2b3C4) (4152), and 
Dy, = (410203) (304) + (d2b3C4) (103) - 
Now, by 5, (4,02¢3) = (dbz) ¢3 — (G03) Co + (A203) C1 5 
and (dyb3C4) = (Azb3) Cy — (2,4) Cz + (gq) Co. 
Substituting, 
D = (db4) [(a, 02) €3 — (4,53) C2 + (A2b3) G1] + (4102) [ (debs) C4 
— (Gy by) C3 + (ag 04) Co] 
= (ayb3) [ (4204) C1 + (41 b2) C4] — [(G2b4) (12s) 
— (,0,) (a304) | Co. 
The second binomial, (ab,) (a,b3) — (a,b2) (a3b4) 


= (yb, — Agb2) (4103) — (1304 — Ags) (be) 
= by [ (a, bs) an) = (ay bo) As | r— 4 [ (ay bs) bs —— (a, bo) bs | 
= b,[ (a,03 — 3b) dg — (A dg —Ayb,) Ag | 
<< 4 [ (a, bs a Cs b,) by ae (a, 0, — (ly b;) bs | 
= Cy Dg (dy bs — Os b.) ee 4 by (dy b3 —— Cs by) 
= (4,04) (yds) Pt Sere wer «5 
eae D = (dy Ds) [ (ay by) C4 a (ay b,) Cog ae (04) Gy |. 
Substituting the expansions of (a,becs3) and of (d2b3¢,) in D, 
we have 
D, = (agb4) [(a, 52) C3 — (1 Bs) Co + (bg) C,] + (103) [( edz) Cy | 
— (04) C3 + (Ag D4) Co | 
= (abs) [(asb4) + (03) C4] — [(arb3) (ads) — (be) (gs) ]Cs- 
Here we notice that the binomial factor of the second term is 
the same as the binomial factor in the last term of D: hence 
equation (K) above, is (a)b3) (a, 04). 
Dy = (dgb3) [(ab3) Cy — (404) C3 + (303), ]. 


Substituting the values of D and D, just obtained, in C, 
we have 


(K) 


\ 


PRELIMINARY NOTIONS AND DEFINITIONS. T 


C= (debs) [ (ay D203) dy — (ty g0,)dy — $ (ay d2) Cy — (D4) C2 
+ (dod4) C$ dz + $ (a, 03) ey — (G04) C3 + (34) C$ de ] 
= (dgb3) (dy by 63) dy — (0204) dg + (a, D304) dy — (a d3¢,) dy]. 


From this value of C the absolute term of VI is obviously 
(ys) [(d,b2C3) Ms — (C4) Mz + (40364) Mz — (AgbgC4) My ]- 


Now the quantity within brackets in C (and in the absolute 
term) of VI is here seen to be composed of four terms, each 
of which contains a factor which is a determinant of the third 
order. We shall presently show that this quantity is a deter- 
minant of the fourth order, and will therefore write, in accord- 
ance with the notation already exemplified, for determinants of 
lower orders : 


(ayboC3) dy — (Ayb204) dg + (Ay D3¢4) dg — (Aad 9¢,) d, = (aydoesdy)  (R). 
Now, 5, (dy B23) = (4402) C3 — (0,03) Cg + (A203) C 5 

* (Gy 0204) = (00) Cy — (04) Cg + (ng) G 5 

(a, b3C,) = (a3) Cy — (D4) C3 + (304) Gy § 

(d,b3C,) = (dgbg) Cy — (ed4) Cg + (Hg D4) Co. 


Expanding the determinants of the second order in the 
second members of these equations according to 4, and sub- 
stituting in equation (R), there results : 


(Ab oCay) = AyD 9C3C4 — gd C34 — Ay D3Coly + gD 1Codg + AghsCydy — Agbocy A, 
ae a, b.C,d, he Cb) C4A ok AD 4CoAl, CTS 40 1Co oY yD 40, A — AoC As 
a 0,04, es (30 1C4Ay eG yD 40515 a 401 CsAy a Asb4C, Ay aaeees A030 do 


— Ag oC4dy + AgdoC4dy + gb Co — Abela — Agb4Cod, + A4d5C.d). 


This expansion contains 4-3-2 = 24 terms, involving 4° = 16 
quantities. Each term contains only one a (coefficient of @), 
one 0 (coefticient of 7), one ¢ (coefficient of z), one d (coefficient 
of t), and contains all the subscripts; a different permutation 
of the subscripts belonging to each term. As before, we find 
that the number of inversions of order of the subscripts is an 
even number in the positive terms, and is an odd number in 


8 THEORY OF DETERMINANTS. 


the negative terms. Moreover, the number of terms is exactly 
the number of permutations of the first four natural numbers. 
Such a function is a determinant of the fourth order, and is 
accordingly designated by (a,b,c,d,). Introducing this nota- 
tion, and dividing by (a,b3;), equation VI becomes 


(a, Do Cs d,)t = (ay bs C34) e VII. 


It is to be noticed that equation (R) of the present article 
sives the expansion of a determinant of the fourth order. 


7. We have now shown how determinants of the second, 
third, and fourth orders arise in the solution of simple simul- 
taneous equations. From the reductions of 6, it is obvious 
that to continue the present method would very soon imply 
difficulties in the simplifications practically insurmountable when 
we attempt to produce determinants of the higher orders. For 
determinants of the fifth order, the process of reduction would 
be found very tedious. Hence, to investigate the properties of 
determinants of the nth order, we are forced to take a new 
starting-point; and in Chapter II. we proceed upon a plan 
somewhat different from that hitherto adopted. 


Values of the Unknown Quantities. 


: A, byC3™M : 
8. From equation VII, 6, t= ene Had the equations 


of set I been so arranged that z should be the last unknown 

i . bod 

in each equation, we would evidently have 2 = eee , 
(ay bo d3¢4) 


_ (Mdy¢3m4) (dy by 374) 
In the same way, ¥ = (a dac,b,) ? v= ‘(bec 


9. Among the many properties of determinants to be estab- 
lished, we may here produce the following theorem, which is 
among the most important of the elementary theorems in the 
subject : 

The interchange of two letters, or of two subscripts, the others 
remaining undisturbed, changes the sign but not the magnitude 
of a determinant. i 


VALUES OF THE UNKNOWN QUANTITIES. 9 


Ist. For determinants of the second order. 


(a) The interchange of two letters. 
(ab) = a,b.—uyb,. In this, if we interchange a and b, the 
second member becomes 


bi Ay i by Ay —— (a,b, oe (hy Dy) re CO) — ioe (a,b) . 


(b) The interchange of two subscripts. 
(by) =,b,—dyb;. If the subscripts are interchanged, the 
second member becomes 
Ay dy — A 0g = — (a, 02 — Anbj) «*. (gd;) = — (a, d2). 


2d. For determinants of the third order. 


(a) The interchange of two letters. 

(0 b,¢3) = (ab) €3 — (4,03) Co + (2b3)¢,. In this, if we inter- 
change a and 0, the proposition is obvious from the first part 
of the demonstration, Ist, (a). 

We have therefore to show that the proposition holds for } 
andc. We have, 5, 


(a2) (dg€3) — (debs) (Ay Cy.) = Ao (1096s). 


In this expression, interchanging b and c, the first member 
becomes (@¢:) (203) — (dees) (402). Since Gy remains un- 
changed, (a, ¢.b;) = — (dbe¢s). 


(b) The interchange of two subscripts. 

(1 b2C3) = (02) 3 — (A, d3) Cg + (dads). (La). If the sub- 
scripts 2 and 3 are interchanged, the second member becomes 
(03) Cy — (b2)€3 + (gb). Since (d3be) = — (d2b3), Ist, 
(b), the second number of (L) becomes 

— (Ab) ¢3 + (4,03) Cg — (Aab3) Cy 
are (bg Cy) = (ay Osta): 
In the same manner it may be shown that the interchange of 


any other two subscripts in (L) changes the sign of the second 
member, .’. the proposition. 


10 THEORY OF DETERMINANTS. 


3d. For determinants of the fourth order. 
(a) The interchange of two letters. 
(ay by Co d4) = (ay by Cat) d,— (a, Ds C4) d+ (a, b C4) de Te (dy bs C4) dy, (L). 


krom 2d, (a), the proposition is obvious for an interchange 
of the first three letters. To show that the proposition holds 
for c and d, we have, 6, 


(ay b,C3) (db3d4) —— (Gz D3 C4) (a by ds) — (dys) (a, by C34) » 
The interchange of ¢ and d transforms the minuend into sub- 
trahend, and the subtrahend into minuend, in the first member. 
Hence, as (a,0;) remains unchanged, (d,.d3¢,) = — (a d2¢3 4). 
(b) The interchange of two subscripts. 
(yb oC3Cl4) ae (A093) dy, eas (ab0¢4) dls + (ab 5C4) dy Le (GybgC4) dy. (M) . 


In this, if we interchange the subscripts 2 and 3, the second 
member of (M) becomes 


(dy bs Cy) dg — (0 b3 04) dg + (A, B24) dg + (3 D204) dh. 


Now, by 2d, (b), (a, 03¢2) = — (A, 2¢3) ; and (a3 b2¢,) = — (Ag bg C4). 
Hence the second member of (M) may be written 


— (0 b903) dy + (0264) ds — (4, bg) dy + (d2b3¢4) dh, 
and therefore (a, bs Cyd) = — (1 bo Cg.d4). 


In a similar manner the proposition may be established for 
the interchange of any other two subscripts. 

It is obvious that two consecutive interchanges will leave the 
determinant unaltered either in sign or magnitude. Notice 
that an interchange of two letters corresponds to a uniform 
change in the order of succession of the unknown quantities in 
the original set of equations. Also, that an interchange of two 
subscripts corresponds to changing the order of the equations. 


10. Applying the proposition of the preceding article to the 

values of x, y, z, and t, obtained in 8, we have 
ee (7 b2C3d4) y= (dy Mo C3 dy) A (a byMs4) | __ (G4 be03 m4) 
(A, b2C3 a4) ” (Gy b2C3 dy) ” (Ay boC3dy)> (a bo 30s) 


NOTATION. 11 


Notice that the common denominator in these values is the 
determinant of the fourth order, formed from the coefficients of 
the unknown quantities. Also, that the numerator of the value 
of x is obtained by changing the a of the denominator into m. 
The numerator of the value of y is likewise obtained by chang- 
ing the 6 of the denominator into m, and that the numerators 
of the values of z and ¢ are similarly obtained by changing the 
ec and d into m respectively. 


Notation. 


11. We have seen that a determinant of the second order 
contains 2?=4 quantities, a determinant of the third order 
3? = 9 quantities, and a determinant of the fourth order 47 = 16 
quantities. It is customary to employ the notation introduced 
by Cayley, and write these determinants so that the quantities. 
(called elements) entering into the determinant appear arranged. 
in the form of a square, with a vertical line on each side. 


Thus (4,52) = |@,0;|; (4 02¢3) =| a,b, ¢| 5 and (a,b,c,4,) =| a,b, ¢, dy] 
Ay Do (bg Dy Cy (lg Dy Cp Ao} 

az D3 Cs G3 D3 C3 ds 

CRUE 


Other forms of notation are also | a, b.| for (a, b,); | a6; | 
for (db, ¢3); | a,b, ¢3dy| for (a, by cz dy). 

There are still others to be described later. In Cayley’s 
notation the elements are so arranged that, regarded as coefti- 
cients of the unknowns in the original set of equations, they 
occur in rows and columns in the regular order in which they 
are found in these original equations. Further, comparing the 
expansions with the square arrangement, we notice that each 
term contains one, and only one, element from each row and 
column, and that there is no other element from the same row 
and column in the same term. Hence, as already exemplified, 
there can be only 2, 3, or 4 elements in each term, according 
as the determinant is of the second, third, or fourth order. 
It will be noticed that the quantities occurring in the abbreviated 


iB THEORY OF DETERMINANTS. 


forms (a, 0,03), (G; 2), |d,bo¢3 dy|, ete., are those found in one 
of the diagonals in the square arrangement, viz., the diagonal 
extending from the upper left-hand corner to the lower right- 
hand corner. This diagonal is called the principal diagonal. 
Similarly, that diagonal extending from the lower left-hand 
corner to the upper right-hand corner is the secondary diagonal. 
Any line parallel to these (principal or secondary) is a minor 
diagonal. Any of the expansions heretofore given show that 
the product of the elements of the principal diagonal is a posi- 
tive term of the determinant. This term being composed of 
the elements of the principal diagonal, is called the principal 
term. The other terms can be formed from the principal term 
by making all the possible permutations of the subscripts and 
prefixing the proper sign to each permutation (5 and footnote ; 
also 6). 

Observe that the order of the letters in the abbreviated forms 
of notation is the order of the columns in the square arrange- 
ment, and that the order of the subscripts gives the order of 
the rows. Thus, |a,b,c;| means the determinant whose first 
column consists of a’s, second column of 0’s, and third column 
of c’s, and that the subscript of each letter in the first row is 4, 
and in the second each letter has the subscript 2, and in the 
third each letter has the subscript s. 


Illustrations are: 


| ds Do Cy| =| Ay ds Ce |. 
Ce De Co 
O,0gC, 


| dz by Cs dy | = | dg b3¢5d3|3 (A, Cod3) =| a, ¢,0,|. 


4 by C4 dy As Co De 
ds Ds Cs Cs (lz C3 Ds 


|a, b, Cy, dy| =| a, 


NOTATION. 18 


The expansion of determinants of the second and third orders. 


12. Though we have already given the expansion of deter- 
minants of the second and third order several times, it will be 
useful here to compare these expansions with the square 
arrangement once more. Also, we are now prepared for a 
convenient mnemonic rule for the expansion of a determinant 
of the third order, to be given in 15. 


13. Since 


at, 0, 
Cy Oy 


= d, bz — dy by, it is obvious that the expan- 


sion of a determinant of the second order is obtained by taking 
the product of the elements of the principal diagonal and the 
product of the elements in the secondary diagonal, and sub- 
tracting the second product from the first. 


14. We have repeatedly shown that 


Gy by Cy | = | a, d, | Cg — | Ay Dy-| Co. + | Ay Dg | Cy. 
A Do Ca Cy Dy ds-Ds (ls Ds 
dts Dg Cy 


From this it appears that a determinant of the third order can 
be decomposed into determinants of the second order, each 
multiplied by the elements in order of the last column, begin- 
ning with the last element. Since any column may be made 
the last, 9, the assertion just made amounts to saying that 
a determinant of the third order may be expressed in terms 
of determinants of the second order and the elements of any 
column. 

The reader will readily see how the determinant factors of 
the expansion in the present article are obtained from the 
original determinant. For example, the cofactor of ¢, is ob- 
tained by striking out the row and column in which ¢, is found, 
and regarding what is left as a determinant of the second order. 


Thus, ' 
a b, 


“Gigs -Dy-- Ey 
a3" Ds C3 


14 THEORY OF DETERMINANTS. 


15. The following convenient rule for the complete expan- 
sion of a determinant of the third order is indicated in the 
accompanying diagram, and is described as follows : — 

The terms composed of ele- 
ments of the principal diagonal 
and of the minor diagonals 
parallel to it are positive, while 
those formed of elements in 
the secondary diagonal and the 
minor diagonals parallel to it 
are negative. The elements 
pierced by the double lines 
compose the positive terms. 
The elements pierced by the 
single lines similarly consti- 
tute the negative terms. In accordance with these directions, 
the expansion of 


a, 0, ¢, 
Cy Dy Cy 
As bg Cs 


IS A, 09l3 + Agb30) + A301) Cy — AgbyC) — Ab) C3 — Ay dso. 


This is identical with the expansion already obtained in 5, as 
it should be. 


EXAMPLES. 
1. Find the values of : 

—4 —6|; |25 18}; |a-b|; |a+6 6]; |a —b bd]; 
Se aes} AQ 75 ba atba b —ce 

1 

~10'—6 | a 7s Pao seers 

8 —8 D0 0 1 

a1 +- 

a 


2. Write in determinant form: — 


7;9;16; —133; ay—ay; Bu — Thy bd; S—2; 3 gh — xy. 
a 


(Suggestion: —7=3 x 2—(1x — 1)=| : 2 Numberless 


other forms could, of course, be given for the same quantity.) 


EXAMPLES. 15 


3. Without passing from the determinant notation, show 
what relation exists between 


a, 6,;| and |b;a,|. Also| w yj and(|mn|. (9.) 
Cy Dy Do Gy mn uy 
4. Compare a b| and ja c|. 
eid bd 
Also compare ao}, \8a 36], and: |3da 64. 
cad Coa 3c d 


5. Write the expansion of the following determinants : 
(dy bs) (a Dx) 5 | Oy Dy, Gn 13 | Gy b Ce | 3 (3 bs C1) 3 | by C3 Cy |. 


6. Find the values of: 


Memos, O1;1000); 1/0 0Dalsla 0 cls |a.be€ 
456 03 6 060 teas) b 0b 06.4 
moo 059 Cae 00) Cit 6 © -O-6 
7. Compare a0cj|and/a0 0 
dof GN see f 
g 0k ghk 
Also compare ambc|and la bc}. 
d me f iret 
g mh k ghk 
Also compare G, 0; -¢, | ANG || .d, Ge a, |. 
ds Ds Cs Cy Cy Cg 


8. State the probable theorems exemplified by the results in 


123 ey 6 
9. Find the value of « in the equations : 
2a ata) pee a OS 
| 5 —6 38 —2 
Bie Te 
(3) 1 [Sac py een —|bbex. 
auc aea b xb 
bb«x aa xb b 


16 THEORY OF DETERMINANTS. 


10. Find the complete expansion of 


a, b, ¢, d, | =|,b,¢, d|. (6, equation (R), e seg.) 
On DT GaOy 
An bn Cr dn 
a,.0, € Ge 


11. Write in determinant form, square notation : 
(1) bfg + etd + heck — hfd — ecg — bik. 
(2) My No73 — My Ngo MeNz1) — My Nz 13+ Mz N21 12— MgNgIy. 


(3) 3ayz —x—y — 2. 


12. Employ 9 to compare the following: 


aobe|and|defi;|m no} and|o 1” mas 
1 el Oe ghk Den 7g 
ghk abc stu ut s 

mnol|and|jomn\}. 

DF F eter igs 

Set u ust 


13. Expand the following in terms of determinants of the 
second order and the elements of any column (14). Verify 
the results by making use of the rule in 15: 


Ct 0 Cy |5 | Lo Yo My | 5 | My Dg Cy |. 
(Cg Dy Cy Ls Yo Nz As bs Cs 
Cz Bg C3 | | &y Y4 M4 | | Ag Og Cg 


14. Count the inversions of order in 


(a) 1854267 
(6) 2 97161, 4°57 
(c) Goi eee ree 
(d) 789658421 
(e) 9876543821 


CARTER he 
GENERAL PROPERTIES OF DETERMINANTS. 
Notation and Definition. 


16. The investigations of the preceding chapter have revealed 
the fact that a determinant of the second, third, or fourth order 
is a function of 2’, 3°, or 4° quantities respectively, and have also 
established a uniform law of formation for these functions. In 
order therefore to investigate the properties of Determinants in 
general, we have but to consider a function of n? quantities 
whose law of formation is given in the following definition. 


17. Derinition. — A Determinant is always a function of n? 
quantities. These quantities, called elements@eing arranged 
in the form of a square consisting of m rows, and thus also of 
n columns, ” quantities in each row and in each column, the 
determinant of these n” quantities is the sum of the terms 
formed as follows: * Each term is the product of n elements, 
so chosen that there is one element from each row and one 
from each column, — but two elements from the same row or 
column must never occur in any one term. The sign-factor of 
each term is (—1)’*%, in which ” is the number of inver-, 
sions of order} of the rows, and % 1s the number of inversions 
of order of the columns, from which the elements composing 
the term have been chosen. 


Note. — Each term being composed of n factors, the deter- 
minant is said to be of the nth order or degree. 


* 22 et seq. will show that the law of formation given in this definition 
is the same as that already observed in determinants of the 2d, 3d, and 
4th orders (3 to 6 inclusive). 

t 5, footnote on inversions of order. 


18 THEORY OF DETERMINANTS. 


18. To expand |a b c| by the definition, we may select any 
def 
mn O 


row, as, for instance, the second row, and using each element * 
of that row in turn, according to the directions given, we shall 
form all the terms of the determinant. For the first term, then, 
taking d as the first element, we see that we can take 6 and o 


for the other factors of a term, and no more, since we have 
then choser one element from each row and one from each 
column, and no two elements are from the same row or column. 
We now have the term dbo. To form another term containing 
d; we can evidently take n and c, giving the term dnc, which as 
before contains an element from each row and column, and no 
two elements from the same row or column. No other 
terms containing d can be formed. The terms containing e are 
in the same way eao and mec; the diagram will sufficiently 
explain the manner of obtaining these terms. 


he ee 
min Oo 
The terms containing f are likewise naf and fbm. 


abe 
-d--e-F 
mn o 


To fix the signs of these terms, we will write under each 
term the numbers giving the rows and the numbers giving the 


* There is a difference in the nomenclature. What we have called 
elements some authors call constituents, and an element is a term. 


GENERAL PROPERTIES OF DETERMINANTS. 19 


columns from which the elements have been taken, and opposite 
each series the number of inversions. Thus: 


dbo dnc eao mec nat fom 

mows 213-—1 231—2 213—1 3821—38 812—2 213-6] 
Columns 123—0 123—0 213—1 123—0 213—1 3821—38 
The sum of the inversions of order in rows and columns of the 
first term is unity; ..(—1)?=—1, and dbo is negative. In 
dnc the sum of inversions of order in rows and columns is 2; 
~.(—1)?=1, and dnc is positive. Similarly for the other 
terms. Affecting the terms with their proper signs, 


a b c|= —dbo+ dne+ eao — mec — naf + fbm. 


ie 
Mm nN O 

Scnotium. — This illustration is inserted only to give the 
reader a clear idea of the meaning of the definition, and not 
because we really employ the definition in the practical expan- 
sion of determinants. In fact, the great beauty of the deter- 
minant notation is that we are able to conduct most of our 
investigations with the help of determinants without requiring 
the expansions at all. “In case it becomes necessary to expand 
a determinant, we have several excellent methods to be given 
later. One method for the expansion of a determinant of the 
third order has been given already (15). 


19. In accordance with the notation already exemplified in 
Chapter I., a determinant of the nth order is written 
Cinoiiae. <euk 
Ga Gye Gein lle 
Gigi lastgess th 
ak fe ae 
This form is shortened to (a,0,¢;...0,) or |a,02¢3...0,|, or to 
> + a,b.c,...1,. In each of these shortened forms those ele- 
ments occur which occupy the principal diagonal* in the square 
arrangement. The form 3+ a,0,¢,.../, is suggestive of the 
manner in which the function is formed. The = + stands for 


oa a 


20 THEORY OF DETERMINANTS. 


the sum of all the terms that can be formed from the principal 
term by permuting the subscripts and prefixing the proper sign 
to each. (23.) 

Another and very convenient notation is obtained by employ- 
ing a single letter affected with two subscripts; the first sub- 
script giving the row, and the second subscript the column, in 
which the element occurs. Thus: 

Qh Any Ohg i «0 Oqy+|- 


C9] Cho Clog eee Clon 
Ons ag Cag na Cee 


PRE GO BRETE Pir tee 
This form may, like the first, be shortened to | dy dy... Ann|, 
(Ay, Cag Ugg +++ Ann) OF B Ay Agy Mgg +. Ann. It may also be still 
further abbreviated to | a, |. A modification of this notation, 
with the two subscripts, consists in omitting the letter alto- 


gether, and writing the determinant thus: 


(1,1) (1,2) (1,8) ... (1,2) | or | 11 1290S e 
C2E 1) AC2 i) {253 \P sets vy) 21 22 23 32% 
CSO 0 8 ED ie aca is | 31 32 33... 3n 
iy Gnd) GiBY iGo ee 
or, finally, /123...n\. 
NS hae 


These last three forms are called the wmbral notation. 


20. The following corollaries flow from the definition in 17. 
They are obvious upon a moment’s reflection. 

Cor. I.—The principal term is always positive. 

Cor. Il. —If each element of a row or of a column is zero, 
the determinant vanishes. 


General Properties. 


21. THEOREM. — Jf in a series of integers which are all 
different, any two are interchanged, the others remaining undis- 
turbed, the number of inversions of order is thereby increased or 
diminished by an odd number. | 


GENERAL PROPERTIES OF DETERMINANTS. 21 


Let the series of integers be Ae Bf C, in which A is used to 
denote the series ay« ... preceding e, B denotes the series hg/ ... 
between e and /, and C the series following /f. 

In the first place, it is evident that if any two adjacent 
integers are interchanged, the number of inversions of order 
is thereby increased or diminished by unity. For let vm be 
any two adjacent integers in a series. If we write mv, we 
introduce one inversion of order if m>v. Or, if m<v, we 
have lost an inversion. Now, since this change cannot affect 
the rest of the series, we have increased or diminished the total 
number of inversions in the series by unity. 

Again, in order to interchange e in Ae Bf C, with f separated 
from e by k, intervening elements, we may first interchange e 
with the elements to the right in regular succession / +1 times ; 
this brings e into the place at first occupied by f. Then, in 
order to transfer f to the place formerly occupied by e, we have 
to pass f over & elements to the left. Altogether, we have 
changed the number of inversions of order from odd to even, 
or from even to odd, 2k+1 (an odd number) of times. Hence 
the proposition. 


22. THrorem. — The number of terms in a determinant of 
the nth order is 1-2+3+-...n=n! 


The simplest way to form the terms of a determinant accord- 
ing to the definition, is to choose the elements from the columns 
in order; that is, the first element of a term from the first 
column, the second element from the second column, ete. 
Choosing the elements in this way, we may take the first ele- 
ment of a term from the jirst column and third row, say, the 
next element from the second column and any row except the 
third, the next element from the third column and any row 
except those already selected, and so on, until all the columns 
and rows have been drawn upon. The numbers of the rows 
from which the elements are chosen will constitute_a_permu- 
tation of the numbers 1, 2, 3,... , and it is (@bviows Yhat 


Arh THEORY OF DETERMINANTS. 


many different ways as there are permutations of the first n 
numbers, that is n! There are. accordingly n! different terms. 


23. Cor. I.— The terms of a determinant | a,b,c; ... 1, | 
may all be obtained by keeping the letters in alphabetical order 
(7.e., choosing the elements for each term from the columns in 
order), making all the possible permutations of the subscripts, 
and prefixing the sign + or — to each permutation, according 
as the number of inversions of order is even or odd. Since 
the expansion of a determinant in accordance with the definition 
would also be obtained by keeping the rows in order, and 
choosing the elements from the columns in all possible ways, 
all the terms of | a,b,c; ... J, | can be formed by permuting the 
letters, keeping the subscripts in order, and prefixing the sign 
+ or — to each permutation, according as the number of in- 
versions of the letters is even or odd. 


24. Cor. II. — Similarly, the terms of | a,,| can be formed 
by making all the possible permutations of the first set of sub- 
scripts and keeping the second set in order; or the terms may 
be obtained by making all the possible permutations of the 
second set and leaving the first set in order. 

Illustrations: To expand | a,6,¢,|, we may write the permu- 

Ohe Oo Co 

As Ds Ce 
tations of the subscripts in a column, and indicate the number 
of inversions of order in each by a figure placed at the right ; 
or we may write the permutations of the letters in the same 
way. Thus: 


132 See cpr 
ae eta 0 t Ob Gt ea 
rae) a pee’ FERS TAS gy na Sp 
Sogn! ee OG sy 2 
ree bigk. CAL OF eee 
PA TES SORA | C500 nea 


The two expansions are accordingly 


~ 


Cy Oa.C3 — Ay Bg Cy + Az Dy Cg — 1g. D9Cy + Ag Dg Cy — 2D, Cs, 
(ty D9 Cs ae Ay Cob, car: Dy Ay Cs ok Dy, Co As a C1 An ds cea C1 Oo Us. 


ee 
' if 
cra? 


GENERAL PROPERTIES OF DETERMINANTS. oe 


To expand |) Gs d33] according to Cor. II, we have simply 
to write the elements for each term with one set of subscripts 
in order; thus, 

Cy Che Gg, Ay Ag Mg, Ay Ag Mz, Ay Aggy My Ay Ag, Ah Ae As 5 
and then for every term, according as we choose from columns 
or rows in order, write one permutation of the numbers 1, 2, 3, 
before or after the subscripts already written, obtaining 
(yy Clos Ags a yy Cs9 Clog aa Cs Cp Clog re C31 Clog Cy2 + Co Cs C43 AR, C91 Che Choo 
or 

yy Coo Clas = Cy Clog Cs9 ee C9 Cy) C33 +- C19 Clog (bz) an C42 Coy Aso Pat C43 C99 Cz}. 

25. THrorEM. — Jn any determinant, if the rows in order 
are made the columns in order, the determinant is unchanged. 


? 
The theorem is an@bvious consequence of 23 and 24. The 


following proof is based directly upon the definition. Consider 
the determinants A and A’, which differ only by making the 
rows of the one the columns of the other. Every term of A 
‘contains an element from each row and column of A; hence it 
contains an element from each column and row of A’, and is 
therefore, disregarding the sign, also a term of A’. Similarly, 
every term of A’ must be a term of A. We have now to show 
that the signs of corresponding terms are alike. Let the num- 
bers of the rows and columns for a term of A be 

9 ory 7-05 sen LOT the rows ; 

1, 50, S$, M,°... Lor the columns. 
Then, by hypothesis, the numbers of the rows and columns of 
the corresponding term from A'will be 

Tt, Oy 8) ys se~ LOr the’ Tows ; 

a, ¥, 6; 7, 0, --- for the columns. 
The two terms (obviously) have the same sign. Hence the 
proposition. =— 


Illustrations : ¢ 
yy Aye Ag Ay | =| Ay Ag As, Ag | 5 | M0, Cy dy | =| Ay Ae Ag Ay]. 
Ala} Chen Mog Mog Any Aes Ago Cag GaOs Cy C5 Dy by bg D4 
Ag M32 Ugg Usa Cy3 Clog Ugg Clas tz Ds C3 dy Cy Cg Cz Uy 


Gy Cas Osg Oss ys Cog Any Ona Geb, Cy Ay d, dy ds dy 


lf 


24 THEORY OF DETERMINANTS. 


26. Turorem. — In any determinant the number of positive 
terms equals the number of negative terms. 


By 23 all the terms of a determinant can be formed by keep- 
ing the letters in order, and making all the possible permuta- 
tions of the subscripts (or 24, case of the double subscripts, 


by keeping one set in order and permuting the other set). We 
n! oo n! 
have to show, therefore, that 5 permutations are even* and 15) 


are odd.* Let 2 and y be the number of even and odd per- 
mutations respectively; then e+y=n! If we interchange 
any two subscripts in each of the # even permutations and in 
ach of the y odd permutations, the even permutations become 
odd and the odd even. Since by the interchange of two sub- 
scripts we could only reproduce permutations all different from 
each other, and already found in the original set of permuta- 
tions, it follows that v= y. 


27. Turorem. — If two parallel lines (rows or columns) of 
a determinant are interchanged, the sign of the determinant is 
changed, but its numerical value is unchanged. 


Let A be the given determinant and A’ the same determinant 
after the Ath and rth rows have been interchanged. Then 
—A=A', 

Let T= + Ad, Bm,C be a term of A, in which A, B, and C 
denote the product of elements from all the rows and columns 
except the dth column and Ath row, and the mth column and 
rth row. Then 7’ (disregarding the sign) is also a term of A’, 
for it contains an element from each row and column of A’. 
Now 7, regarded as a term of A', contains exactly the same 
inversions of the columns as it does when regarded as a term 
of A; but the number of inversions in 7, as to rows, when 
considered as a term of A’, is an odd number, more or less, 
than when considered as a term of A. For, in writing the 
numbers of the rows, to determine the inversions, we write 


* This language, of course, signifies permutations in which the number 
of inversions of order is even or odd respectively. 


GENERAL PROPERTIES OF DETERMINANTS. 25 


them just as we would for A, except that k and r will have 
changed places (d, being found in the rth row, and m, in the 
kth row of A’). Thus every term of A is found with the oppo- 
site sign in A’,...—A=A’. By 25 the proposition must be 
equally true for an interchange of two columns. 


Illustrations: 


— | a, dy | = — [a doc3 + a2 b3¢, + gb, 0 ap by G te 

Gs Da.¢ ani} 

et ae — A390, — yb, C3 — A, 0369] = Og Os Os 
Ga Ds Ce Cz Do Cy 
Cy Dy Cy dy — Cy b, Gy dy | — Cy b, Cy a, = -_ Oy dy Cy dD, ie 
Cs by Cy dl, Cl by Co d, / Cs bs Cy dls Cbs ds Co bs 
As Ds C3 ds Oy Og Cy dy | Ob, Og Cy Oy ly dy Cy Dy | 
4 Dg C4 dl, As bs Cz dls ar) by Cy dy ae) de Cy by / 


(a, by 34) —— 1 re (dy bi Cs dy) = (yb; Cy d,) —— (Cy bs C4d,) St (ds bo, d,). 


28. Cor. — If two parallel’lines of a determinant are iden- 
tical, the determinant vanishes. 

For, by the proposition, if the two identical rows or columns 
are interchanged, the sign of the determinant is changed. But 
the interchange of two identical lines cannot affect the deter- 
minant. ‘Therefore 


A=—A, 
Pi ®. eae 0, ora =U; 
Illustrations : 
abe|=aec+dbe + abf— aec —dbe —abf=0. 
def 
abe 
Cy Dd, Cy dy — Oy Og Ag C4 — O. 
Ge Oo Ca Ae C,Co CpeOs 
Og Og Cy Uy dl, dy dy As 
(Q, bo Cy dy) = 0. | dy by ag dy | = 0. 


29. If in a series of integers, 


fy ted, €, tn 73 


_/ 


26 THEORY OF DETERMINANTS. 


the first is passed over all the others in succession to become the 
last, the others remaining undisturbed, thus, 
Cis Mba eCuls IRS Nai hs 

the numbers are said to have been cyclically interchanged. It 
is obvious that a cyclical permutation of n given numbers 
can always be effected by n —1 interchanges of two adjacent 
numbers. Accordingly, a permutation containing an odd or 
even number of inversions still contains, after a cyclical inter- 
change, an odd or even number of inversions if » is odd; if n 
is even, however, a permutation containing an odd or even 
number of inversions will, after a cyclical interchange, contain 
an even or odd number of inversions respectively. 

From a given permutation of n integers any other permuta- 
tion can be obtained by cyclical interchanges. Thus, from 


fadcegb 
we get cagfdbe 
as follows : — cfadegod 
cafdegb 
cagfdeb 


cagfdbe 


The groups in which the cyclical interchanges take place are, 
of course, fadc, fa, fdeg, f, d, eb. 


30. The previous article (or 27) establishes the following 
theorem : 

THEOREM. —If in a determinant A any row or column be 
passed over k rows or columns in succession, and the Bae 
determinant be denoted . A’, then 


=({— 1)" AG 
Illustrations: 
®y Y1 % by | =| Xs Ys Ms ty | =| Wg tz Yg 23 | = —| MY 2 
Ho Yo Bq ty 1 Yy % ty Uy ty Yy 2 Ly bo Yo Xs 
Us Ys Bs ts Xe Yo Xs ty Ly bo Yo Bo Uy by Ys Ms 
Ly Ys Ry by Ly Ys My by Uy ty Ys Ms Xs ts Ys 23 


|X Ya Ve Ws | = — | WH Yo Vg Wo |= —| Vy Be Ys Wo | =| Li Yo Ws Y |. 


= 
be 


GENERAL PROPERTIES OF DETERMINANTS. 27 
EXAMPLES. 
_—~~1. The student who has not done the examples at the end 


of the first chapter may attend to them before proceeding to 
the following. 


2. What terms of | a, 0,¢,d,| contain b, ds? 
3. Write the terms of (x, 2 ww 32, ts) that contain t, yj, Ws. 
4. Show that in a determinant of the nth order only two 


terms can have (n—2) elements in common, and that these 
terms have opposite signs. 


~ 


5. What is the sign-factor of the term containing the ele- 
ments in the secondary diagonal of a determinant of the nth 
order ? 

6. Show that the sign of a term is independent of the 
arrangement of the elements composing it. 


7. Show that the sign of a determinant is not changed by 
any interchanges of rows and columns that leave the same 
elements in the principal diagonal, whatever the final arrange- 
ment of the elements in this diagonal. 


Sug. If agg ayy daa ++» Max be the final arrangement sought, 
dgg can be brought into the first place by 2(8— 1) interchanges 
of two rows and columns, ete. 

8. A corollary from 30 is: Any element a, can be trans. 
ferred to the first place by making the ‘th row and kth column 


the first row and column, and then multiplying the determinant 
by (—1)***. 


31. THErorem. — Jf every element of any line (row or column) 
is multiplied by any number, the determinant is multiplied by 
that number. 

Since every term of the determinant contains one element, 
and only one, from the line mentioned in the theorem, the truth 
of the proposition i€evidend 


i 


28 THEORY OF DETERMINANTS. 


Illustrations : 
ar or ar| =7r|a,b,¢|=| a rg |=7\a, ae 
Oe De Ae Oy Ce Ay Oo Cy otiea 
Gg by é3 | As Ds Ce Clg Def Cy : 
— be Ce 
Yr 
(3 
— Ds Cs 
\7 
A=lbcaa?|; then abcA =| abc a’ a®|; «. A=|1 We 
cab abe b? B® Lee 
abe ¢ abe ¢? 1 Che 
Let the student show that 
bed a a?a®|=|1 a’? a’ at |. 
cda b b? be 1:67 6° DO 
dab ei ee ich eie 


| abe d a’ d® Led | 


32. Cor. I. — Changing the signs of all the elements of any 
row or column changes the sign of the determinant; for it is 
equivalent to multiplying the determinant by — 1. 


33. Cor. Il. —If two rows or two columns differ only by a 
constant factor, the determinant vanishes. For we may divide 
each element by the constant factor, and write this factor as a 
multiplier before the determinant. Then the determinant van- 


ishes by 28. 


Illustrations : 

ry ett Mg =a”ial 1 |=0. 15 7 |= 54 1a 
Dew 6 baa 2106 226 
Gata? C a? a BLLon 339 


34. THEOREM. — If each element of any line* of a determinant 
isa binomial, the determinant equals the sum of two determinants ; 
the first of which is obtained from the given determinant by sub- 
stituting for the binomial elements the first terms of the binomials, 
and the second determinant is obtained from the given determi- 


* Since it has been shown (25) that what is true of the rows of a 
determinant holds for the columns, it will only be necessary hereafter to 
state a proposition with reference to either rows or columns. 


oo a 


i 


GENERAL PROPERTIES OF DETERMINANTS. 29 


nant by substituting for the binomial elements the second terms of 
the binomials. 


By the definition every term of the determinant must contain 
one of the binomial elements. 


Let (m+n) bghk...l 
be one of the terms of the given determinant; this may be 
written mbghk...l+nbghk... l. 


Now the first term of this sum is a term of the original deter- 
minant, with m written for m-+n, and the second term is a 
term of the original determinant, with n written for m+n. It 
is obvious that a similar statement applies to every term of the 
given determinant; hence the proposition. 


Illustrations : 


Cy + ay by Cy = Cy by, Cy ao ay Dy Cy e 
My + ay De Cy My De Cy dy Dy Cy 
CAs + Ag bs Ce Cs bs Ce Og bs Ce 


“be eal" My Ny = Vy My Vy —— Yi My Ny . 
Y_a— Yo My Nog XY Mz Nog Yo Mog No 
V3 —Y3 Mz Ng Xz Ms Ns Ys, Ms Ns 


35. The preceding theorem is evidently capable of extension. 
The same reasoning applies to a determinant any line of which 
is composed of polynomial elements, or, again, in which each 
element of every line is a polynomial. That is to say: If each 
element of any row is a polynomial of q terms, each element of 
another row a polynomial of r terms, each element of another 
- row a polynomial of s terms, etc., the given determinant is the 
sum of sxqx7r... determinants. Thus: 

atbte m—n tl=la m—n t|+| 6 m—n t 
dt+e—f o+tp u| |d o+p u e o+tp wu 


g—h+k q—r v| |g q-—r v}| |-h q—-r v 
+/ cm—n tl=|a m titla—n t|+| bmet 
ow  O+p' ul jd o uy id p u eo u 
k q—r v| |g g v g—r v| |j-h q v 
+| b—n t i+ em ti+| c—n t\. 
e pu| |-fo wl |-f pu 
—h—r v k qv\| | k—r v 


30 THEORY OF DETERMINANTS. 


36. Reciprocally, If q determinants differ from each other 
only in a single line, the sum of these determinants is a single 
determinant, derived from any of the given determinants by sub- 
stituting for the elements of the line which is different in each 
of the q determinants, the sum of the corresponding elements of 
the q determinants. 


Tilustrations : 
abc\|\+labc|+\|mno\|= al b c : 
def LY % abe d+tx—m e+ty—n f+z—0 
ghk ghk| | ghk gq h k 
The student may show that 
Cy Cy bd, Cy — | Cy by Ly ®) = 0. 
Gs Ay bg Cy gots Com) 
OSs 7 Oa ots Gs Ds Cg “Os 
O meth toe es Cha Os Gy Oe 


37. THreorem. — A determinant remains unchanged if the 


” elements of any line be increased or diminished by equal multiples 


of the corresponding elements of any parallel line. 


We are to show that 


Cy by Cy ad, eee Li = Cy db; G9, £950, +...44 dy eee ly 
eet) Oe 25006 Gy Oy | CoG: Cork Gobet... ly Gel eveme 
pee So sagas eee 2 tg Og | Cg Qi. Gy 2 Gols .. ttl, “Uy eee ne 


dy Dn Cy Uy vse dy] [Oy Oy Cnt G Gy + G2, tel, dy one dy 


Calling the first determinant A, and the second A’, we have, 35, 


A'= Cy Dd, Cy dy, eee # tq Cy bd, Cy ad, eee L 


hs “Dg Gods «2s te Ge Oy Ug Oe ..- by 
On b, Cr dy, meus L, | Gn b, a, d, ef i 

sb Gg| Gp peas peels (ck +| ay Dia, Oy eee 
Clg Deg Os ts yee ies Obs Oy he Ug x 2le 


la, 6 Gees a. Bhd: ae 
Whence, since all the determinants of this series, except the 
first, vanish, A = A’. 


GENERAL PROPERTIES OF DETERMINANTS. 31 


This theorem is of great importance in simplifying and ex- 
panding determinants. Thus: 


1 ab+cl=|1 aa+t+d+e =(a+b+c)) lal|=0. 
1 bc+a 16a+6b+c Tol 
1ca+od 1 ¢ a+b+e 1-64 

@- ad+3a4+6|/=(38(a+1) 8(a+4) 3(44+7)|=0. 


atla+4a4+7 a+il a+4 a+i7 
a+2at+5a+8 a+2 a+é5 a+s 

The second determinant is obtained by adding the second 
and third rows to the first row. 


iets lt) 1/—|—1111|=—|02 2 |=—8/011|/=—16. 
he th 0022 20 2 RUT 
Per seo" 1 0202 220 110 
| ee ee 0220 


The second determinant is obtained by adding the first row to 
each of the others. 

The third determinant is obtained from the second by ob- 
serving that as all the elements, except one, of the first column 
of that determinant are zeros, all the terms vanish that do not 
contain (—1). 


Oa 413) 7 11 4 1|=8)7 ~10:—10/3110 101 
915101 «118-1510 ip —24 16)". |e4 16 
Bo 6 yes eek ie Lp eS 


=80 (16-24) — —240. 


The third determinant is obtained from the second by subtract- 
ing three times the first column from the second column, and 
twice the first column from the third. 


Minor Determinants. 


38. If in a determinant any number of rows and the same 
number of columns are suppressed, the determinant consisting 
of the remaining elements (their relative positions being undis- 
turbed) is called a minor of the given determinant. 

If one row and one column are suppressed, the result is a 
principal minor, or a first minor; if two rows and two columns 


a2 THEORY OF DETERMINANTS. 


have been suppressed, a second minor ; and soon. The elements 
common to the suppressed rows and columns also form a deter- 
minant called the complementary of the minor, formed from the 
rows and columns that were left undisturbed in the original 
determinant. 

Thus, | a 0, Cy ds 
dg bs Cy dy 
| ad, by esd, |; also |a,b.| and |¢,d,|, or b, and |a,c,d,|, are 
complementary minors of |d@,0,¢3;d,|. |e, ds| and | dy by e;| are 
complementary minors of |a,0,¢;d,e;|. In general, if the 
determinant is of the xth order, two complementary minors will 
be of the rth and (n — r)th orders respectively. A determinant 
n?(n—1)? 

(2!)° 


and are complementary minors of 


of the mth order has vn? first minors, second minors, 
ete. 
Since we usually denote a determinant by A, it is convenient 
to denote the minor obtained by suppressing the row and 
column of a3 by A,,; that obtained by suppressing the row 
and column of d, by Aq,, ete. 
Similarly, a second minor, obtained by suppressing the rows 


and columns of 0, and c,, is denoted by Az,c,; and so on. 


Equally efficient notations are: D Ct and D eo. : for the 

: (m, (m rk, 
minor obtained by suppressing the /th row and mth column, 
and the minor obtained by suppressing the /th, nth, and tth 


rows, the mth, rth, and Ath columns respectively of | a,,|. 


39. Since, by definition, every term of a determinant con- 
tains one, and only one element from any line, the determinant 
must be a linear homogenous function of the elements of any 
one row or column. Thus: 


| cy by Co eee Ne [= ty Ay + My Ay + dz-Ay + +: +a, A,, 
=4A,+0,B +4 CO, ++-+h, Ly; 
= (C+ % OQ) + ¢,C,+ +++, OC, 


’ 


in which A,, A,...A,; ©, C,...0,,; ete., denote functions of the 
elements found in the rows and eclumns outside of the particu. 


GENERAL PROPERTIES OF DETERMINANTS. Bo 


lar line, in terms of which the development is given. In the 
next article we shall find the values of these functions. Since 
we may regard the determinant as a function of x’ independent 
quantities, each of the coefficients A,, A,, ..., may be obtained 
by differentiating | a,b. c;...d,| with reference to the quantity — 
whose coefficient is desired. Introducing this concept, the 
equations above written become respectively 


aig qo F ee 


2 3 
dd das Ady, 


dA dA dA dA 
= aq, —_ + 0, —+aq — ae ‘Roane 
; day oe db; : de, a te dl, 


dA dA dA dA 
= C C p= eee paca 
de, dlcy ape des a Ge 


— 


This notation is often employed. 


40. Tureorem. — The coefficient of any element in the expan- 
sion of a determinant is the first minor obtained by suppressing 
the row and column to which the element belongs. This minor 
is taken with the + sign, if the sum of the row and column 
numbers, to which the element belongs, is even; with the — sign, 
if this sum is odd. 


Consider the determinant A= +, 0,¢;...1,, and suppose 

A to be written, 
A = a, Ay + dy-Ag + A3Ag+ +++ +O, A). (1) 

We can collect all the terms of A that contain a, and write 
this element as a factor of the polynomial that results ; we can 
do the same for a, @;, and so on, for each element of the first 
column. These polynomials are A), A,, A;, etc. Now since 
A, is the cofactor of a, it can contain no elements from 
the first row or column; hence a4, can be obtained from 
S+a,b.c...1, by considering a as fixed, and making all the 
possible permutations of the subscripts of the remaining letters, 
i.e., by multiplying a by % + dy C3 --- In» 

Hence, A, = Aq,, the minor obtained by suppressing the first 
row and first column of A. 


34 THEORY OF DETERMINANTS. 


Now, we can bring a, into the first place by one interchange 
of rows: we then have A=—S+a,0,¢,...1,. Employing the 
same reasoning as before, A, must be obtained by multiplying 
ad by —3+b,c,...1,3; whence d,=—Ag,. Again, as can be 
brought into the first place by two interchanges of two rows ; 
whence A;=Ag,, and so on. Finally, a, can be brought into 
the first place by »—1 interchanges of two rows; hence, as 
before, A, = (— 1)”7"Aa,. 

Substituting these values of A, A,, etc., in (1), 

A = a, Aa, — do Aa, + dg Ma, — 12+ + (— 1)", Aaa 

Since the columns may be made rows, it is evident that 

A =) Agq, — 0, An, + 4 Ac, — oe + (—1)”*1, Az. 

It remains to be shown that the proposition holds for an ele- 
ment not in the first row or column, that is, Ay,=(—1)*** Aa,,, 
for the coefficient of the element in the 7th row and Ath column. 
We may transfer the ith row to the first place by i—1 inter- 
changes of two rows, and the Ath column may likewise be made 
the first by k —1 interchanges of two columns. The element 
under consideration is now in the first place. Calling the trans- 
formed determinant A’, we have 

Ae (1) Alor Ae (am 
Whence Ag=(—1)**Ag,. 

41. Cor. I.— A determinant can be developed in terms of 
the elements of any line and their principal minors. The signs 
are alternately + and —:; and the first term is + or —, accord- 
ing as the number of the line is odd or even. 


Illustrations: 
Ay D, Cy d, — at, | Dy Cy ds — Cg D, Ci d, -e As by Cy dy ——— 4 db; Cy dy, 
Oe 05° Co Ae 0; fs ds bs C3 ds| bs Ca Ue by Cote 
Ga Deeeas de Tear by Cy dy| Oj 650 Dy C_'te 
[C4 by Cy Ay 


= —Dy| My C; dy |+0, | ay C3 dy | ~D3|Q, Co dy ]+,] a, co ds| 
—— Cy| Ay bs dy|—cy| a, bs ds | +es|a, bo d,|—cy | ay by ds| 
= — | dy Cy ds |g] dy Cy Az |— Cy] by dz | +g] a De C5] 


GENERAL PROPERTIES OF DETERMINANTS. 35 


42. 41 obviously gives a ready way of expanding any deter- 
minant.* For we may express the given determinant in terms 
of the elements of any line and their principal minors; these 
minors will be determinants of the (n—1)th order. By a 
second application of 41, each of the minors in the first expan- 
sion may be expressed in terms of the elements of any line and 
their principal minors, which minors will be of the (n—2)th 
order. So by successive application of 41, any determinant 
may be expressed in terms of determinants of the second order ; 
and these latter, being binomials, can be at once written out. 
Thus: 


=1(15—16) —2(10—12) +3(8—9) =0. 

| y dy ¢3 dy | =a | bo €3 | —Ag| D, C3dy| + 3| D1 CoAy| —Ag| Dy co s| 

= [b2| ¢3d,| — Bs] Cy dy| +04| Cds] ] — [01 |C34,| —Ds| c, dy| +04| ce, ds] ] 
+ ds | | Coy| —D2| e, dy| +04| do] ] —A4[D1|CoA3| —D9| eds | +03] ¢ do] ] 
hy 0g, — 0, 0,0, d, — 0, De Co dy + 1050, d, 4+ A, 040g dg — 00405, 


— Ag), C34 + Mb, 04d + 


a Obs Dy Cy Uy = 0,0) Cd. ma 
— 00, Cody + O40, C3 dy + Aydo¢,d3 — Ayd9 C3, — yb, dy + Ayd3Cod, 


43. As another corollary from 40, it is evident that if all 
the elements of any row of a determinant except one are 
zeros, the determinant equals this element into its corresponding 
minor, taken with the proper sign. Thus, if the element is in 
the ith row and Ath column, 7.é., d,, then A=(—1)** ay, Aay- 


Illustrations: 
5643 |=—2|64 38 |=—2/] 0—2 —3 |=2|-2 si 
2000 Bar| ra 1 i" 0 
poe tf je i | ok 0 ; 
7 aa 


* Compare 15. + Compare 6. 


36 _ THEORY OF DETERMINANTS. 


The student may establish the following : 


GQ, by 6, Ay |} == O7D, Ce dy. 000d, |=a,0,c,0;: 
0 05 Cy dy 00 ad, 
0. 0. cands 0. Ds yds 
000d, Oy b4-Cy Ay 


44. From the last two examples it appears that if all the 
elements on one side of either diagonal are zeros, the determinant 
reduces to a single term, viz., the term composed of the elements 
in the diagonal which contains no zero elements. 


EXAMPLES. 3152 
1. Show that the following determinant vanishes : : ; : . 
2 Pen ie aa am iy aN Bae ho 615219 
F4 —3 8 2 3 8 
6 5 —9 3.5 9 
a. now. that |.a.* 8) eles 1 1 ses 
! ! EG brane ! ! i 
a fon 2a| ORY eee idea 


This can be readily established by multiplying the columns 
by By, ya, af, respectively, and then dividing the first row by 
aBy. A similar reduction can be effected, in general, whenever it 
is desired to reduce a determinant to one in which the elements 
of one line are units. 


4. Find the expansion of A= 


We notice that 20 is the L.C.M. of the elements in the first 
row; hence, multiplying the columns in order by 5, 10, 4, 2, 
there results 


ees 20202020] j1111 
i Fe See LO 246 5jAG 24h 
85 30 0 10 126 Gee 
O20 2CrE Ose ae 


+ =e 
ae 


_ GENERAL PROPERTIES OF DETERMINANTS. 57 


Now, subtracting four times the first row from the fourth 
row, two times the first row from the third row, and six times 
the first. row from the second row, the last determinant becomes 


eto Li di} —1-4. 18/2 —) Wiis 
—14 18 0 5 4 —2 —3 6 —2 
09 4—2 0 —4 1 1] 0 Tie 1 
—41 1 0 
=—6/ 71 —7|=—6| 64 —7 |=— 384. 
—1 1 | ea | 
AK kat iP ant Os As paeet 1 1 1 vi r 
1 99 3} ~~ Gy dng | 42391 1 A3D2 A, Ags |’ 
Cy oN) Cz Og As Cy Ay As Cg Cy Cy Ce 
also 0 dy a3 |=] 0 Mos La a 
b, by bs Jay te Dake, Os Oy C, 
Cr Gay 1 dg, Cg Ag Dy Cs | 
6. |la a |=(B—y) (y—2) (4-8). 
1B 
ey 


A vanishes if a=, or B=y, or a=y; hence a—f, B—y, 
and a — y must be factors of A. Now the product of the three 
differences is a function of the third degree in a, 8B, y; so is A; 
hence the product of the three differences can differ from A only 
by a constant factor. Comparing the term fy’ (the principal 
term), we see the factor mentioned is + 1. 


7. Show that 


eet = —(8 —y) (a— 8) (y—a) (8 —8) (o—B) (7 — ops 
ey 8 
ae fp? 2 62 
| a fom 7 6° 
Notice that Examples 6 and 7 give in determinant form the 
product of the differences of the roots of an equation whose 


roots are a, f, y, «-- 


Peexpand |S 7 2°20 |s:also} 1 a —£\|. 
oo. tao —a 1 y 
AWAD Bhd i B-y 1 
Bil 01) 6 


* Compare example 3. 


38 THEORY OF DETERMINANTS. 


Expand the first determinant in terms of the elements of the 
third row and their principal minors, since two of these elements 
are zero; then observe that two elements in a row of each of 
the resulting determinants are unity; hence, each determinant 
can be readily reduced to one of the next lower order by 35 
and 42. ’ 


9 |0¢b d|=@?4+0?2+ Cf? — 2 bcef— 2cafd —2abde. 
c 0 a e | 
Diab t Orae 
Ciel FQ 
10. Expand |—-a@ 6 c¢ dj}; also l 25 eae 
pie-OF he 16 —a 1 y' —f' 
e d—a +b —B-y 1 a 
dc 6-—a —y B-a 1 


11. Establish the following identity, and express either 
determinant as the product of four linear factors : 


0101 J aes 
LO ae oe OS an 
Loe A ar 7 2-0-4 
Ly a0) 2-1 1 


12. Simplify thth, ag +h +h, ag th, +he 

Bi thoth, botheth, b3+hotheg 

(+ us +h, coths tk, ¢3+h,+ks 
1 1 


One 
e 


13. Show that|* ytz+t wt+y z+¢t |=0. 
y 2+t+u y+tse t42 
2 t+a+y 2+t w+y 
t wt+ty+2 t+a ytez 
_ 14. Express as a single determinant: 
(1) | ay by ¢5| +] ae By ¢5 | —| ag dy cs}. 
(2) | a be C5 |—| ay Bs 5 | —| ay bs C5] +] ay Be C5} 
15. |q+d,4+d3 dotastay, Ag+ Ag+, Ag+ A, +, 
D+ be+bs bot bs+ by bg+b,+d, bs +0,+ by 
Crt CoA Cy Cot Cyt Cy Cot G+ C+Gte 
d+ d,+ds do+ds+d, dst+d,+d, d,+d,+ds 
= 3 | aydocsd, |. 


Ls 


GENERAL. PROPERTIES OF DETERMINANTS. 39 


x 16. 1 


sin? A 


a? bsin_A esin A 
bsin_A 1 cos A 
csinA cosA 1 


= a? — (0? +c?—2be cos A). 


a+b+ne (n—1l)a (n—1)b |\=n(a+b+e)’. 
(n—l)c b+e+na (n—1)b 
(n—1)e (n—l)a ctatnbd 
“18. | (a+b)? Ce Cc? = 2abe(a+b+c)’. 
al? (6+c)? al” 
b? b? (c+a)? 


19. What is the coefficient of ag, in |a; |? 


20. From the first five rows of | a b,c; dye; fg G7 hg| write all 
the possible minors that can be formed, and their complemen- 
taries. 

How many minors, each a determinant of the Ath order, can 
be formed from any & rows of | q,,| ? 


21. If each of the elements of any line is the sum of the cor- 
responding elements of two or more parallel lines, multiplied 
respectively by constant factors, the determinant vanishes. 


22. Show that 


Geepeni=i lt 0 0 0 }=/| a, 0, ¢ yl=la, b Gq um % |. 
Gy Dg Co #1 Gy 04,6; Oy Dy Co Yo On Oy Co Un Vo 
As bs Cs Bo My Vo Ca Gig! Og: Ca Ys dg Os Cy Ug Y; 
Xe Ms Ds Ce eee OI fe) Oe OF ioe 
Oat cosas 


From this example it appears that any determinant may be 
expressed as a determinant of higher order by writing a zero 
above every column, prefixing a 1 to the row of zeros thus formed, 
and filling in the new column having 1 at the top with any n 
finite quantities. 


23. If in any determinant each element of the first row is 
unity, and if each element of every other row is the sum of the 
elements above and to the left of it in the preceding row, 
commencing with the element directly above, the determinant 
equals 1. 


40) THEORY OF DETERMINANTS. 


24. Any determinant of order n, in which one element is 
zero, is equal to the product of two factors, one of which is a 
determinant of the nth order, in which every other element of 
the row and column containing the zero is unity. 


25. If in any determinant the first element is zero, and if 
each of the remaining elements in the first row and first column 
is unity, the determinant is unchanged when each element of 
the minor corresponding to the zero element is increased or 
diminished by the same quantity. 


26. A determinant of the nth order is expressible as the 
sum of n determinants, the first of which is obtained by chang- 
ing into zero each element of any line except the first element, 
the second by changing into zero the elements of the same line 
except the second element, and so on. 


27. If in two determinants A, A! of the nth order, the first 
row of A is the last row of A’, the second row of A the (n—1)th 
row of A’, the third row of A the (n—2)th row of A’, and so 


: n (n-1) 
on; then ne ep a eek ea, 


28. If in two determinants A,A’ of the nth order, the first 
row of A when reversed is the last row of A’, the second row 
of A when reversed is the (n—1)th row of A’, the third row of 
A when reversed is the (n—2)th row of A’, etc.; then A=A’. 


45. THrorem. — If the elements of any line in a determinant 
are respectively multiplied by the complementary minors taken 
alternately plus and minus (i.e., the co-factors) of the correspond- 
ing elements of any parallel line, the sum of the products is zero. 


Consider the two determinants, 


Desa clahes to Qa. ‘Ug "Cy aes 
Otel ess 0, Oe 6 Aree 


DD ae ane 


iil iia eee Oe DE ee 


GENERAL PROPERTIES OF DETERMINANTS. A] 


where A’ differs from A only in having the kth and pth rows 
identical. Employing the notation of 39, and expanding in 
terms of the elements of the pth row, 

A=4,A,+6,B,+¢,0, ++: +1,2L,; 

A'= 4,4,+6,B, +60, ++ +1,L,=0. 
Comparing these two expansions, we observe that the second 
may be obtained from the first by substituting for the elements 
of the pth row of A the elements of the kth row; that is to 
say. if the elements of the kth row of A are multiplied by the 
co-“sctors of the corresponding elements of the pth row, the 
resuit is A’; since A'=0, the proposition is established. 

Illustrations : 

If in (a,09¢3) = ay (b9¢3) — My (0, C3) + 3 (0,¢.) we multiply 
the elements of the second column respectively by the com- 
plementary minors of a, a, a3, there results 

by (beC3 — 3 C2) — bz (dCs — bgCy) + bs (BD, Co — bee) = 0. 

Let the student prove the proposition, using a determinant 

_ of the fourth order. 


46. A determinant is said to be zero-awial if each element of 
the principal diagonal is zero. Thus the following are zero- 
axial determinants : 


Oye Gar: 02 By. 6, ay. 

Ge DO Cy a 0 G de 

a, 6, 0 ads bs O ds 
a, 04-0, 0 


47. THeorrem. — Any determinant may be decomposed into a 
sum of zero-axial determinants: the first of these is obtuined by 
substituting zero for each element of the principal diagonal of the 
given determinant; the neat n, by multiplying each element of 
the principal diagonal by its complementary minor made zero- 


axial; the neat 5 (m1) by multiplying each product of pairs 


of elements of the principal diagonal by its complementary minor 
made zero-axial, and so on. 


42 THEORY OF DETERMINANTS. 


fond 


In A” = |a,b,c3...1,| change the elements of the principal 
diagorial into zeros, and let AS” denote the resulting determi- 


nant. Whence, 
A® = 10 0 Gq +. G 
Gn WUE Gs rasta 
Og 0g 0) con dy 
Ce Dy Fee zea 
Let A’? denote the minor of Aj” obtained by suppressing 
any one row of A,” ; Aj"? denote the minor obtained by sup- 
pressing any two rows of Aj”; and, in general, let Aj” denote 
the minor obtained by suppressing any 7 rows of Aj”. Also 
let C, denote any product of the elements of the principal 
diagonal of A™ taken 2 and 2; C, any product of those elements 
taken 3 and 3; and, in general, C; any product of the elements 
of the principal diagonal of A” taken 7 and i. Now, Age 
evidently contains all those terms of A“ which involve no 
element from the principal diagonal. (,A," ” must be one of 
those terms of the series which involve only a single element 
from the principal diagonal of A”; consequently PHN se 
will be the sum of all the terms that contain only one element 
from the principal diagonal of A”. Similarly, 3 C,A{" ” will 
be the sum of all the terms that contain only two of those 
elements. And, in general, 3C;, Ae will be the sum of all 
the terms containing 7 elements of the principal diagonal of 


A™, Whence, 
(n) — a ™) ~ (n—1) (n—2) (n—8) (n—) 
AM) — A+ SO Al + 30,AP 9 30.4, 
f+ +3C, Ay + C,. 
It is to be noticed that AY = 0; t.e., there is a break in the 


series, — there being no term containing only n—1 of the 
elements in the principal diagonal. 


Illustration : 


4% 4)/=|0 yx %/+2,|0 2, + Y2/0 2/+2%/0 y% + 2 Yo Zz. 
Xo a 2 Ro Xo 0) Ro U3 0 Xz 0 Xe 0 
1% Ys 2s Xs Ys 


GENERAL PROPERTIES OF DETERMINANTS. 43 


48. Turorem. — If each consecutive pair of elements in the first 
row of a determinant A is taken with each pair of corresponding 
elements of the other consecutive rows to form determinants of 
the second degree, and if these determinants of the second degree 
are used in order as the elements of a new determinant A', then 
A equals A' divided by the product of all the elements except the 
Jirst and last in the first row of A. 

We are to show that 


| hy b, | by Cy ry l, | 
{Gg Oo} | 02 Cg] “" | 1 Ly 
Ay Dy Cy eee Tr) ls pom 1 fr b b z a 1 i 
Mz Dy Cy w+» Ty ly 0,0, d,...7) : a ‘ i > 7 - 
As bs C3 eee Ts 3 d3 3 3 C3 I'3 3 
On, b, Cy see Tn by Cy h, | dy é} r; l, 
alte 
Un Dy | by Cn ln L, 


calling the first determinant A, and the second A’. Multiplying 
the first column of A by —?,, and the second column by q, 
and adding, there results 
—b,A= 0 OG aw Ty oh 

me Ot 0,05 Og Cy... Te le 

— 0, + 0,0; Ds Cy... 13 ts 

at) Yi Pee 7s sd ee ee ee 
Now, multiplying the second column by —«, and the third 
column by 0,, and adding, we have 


3? 
b,¢,4 = 0 0 Cy eee rT) l 
a Cy D, a Cy bo aie bs Cy 4. Dy, Cg Co eee vip) l, 


ri ls Dy _ Cy bs a: bc, + DCs Cz coe Ts l, 
— Ay, b+ Cy b, ec Cy Tug by Cy Cy vee Ty l, 
Proceeding in a similar manner, we have, after (n—1) 
transformations, 
(—1)" 1b, ¢,d, eee LA 
= 0 0 0 eee 0 Lh . 
— dob, + a,b, — bye, + O15 — 6d, + ¢,d,... —Fol, +71, is 
— db, + a,b, — Doe, + dye — od, +d... —7ol, + 7,1, be 


a a,b, -+ ab, =o 8 oso — Cy y+ Cyd, o+ —T,h +t, l, 


44 THEORY OF DETERMINANTS. 


Now, applying 43, and dividing by (—1)"1'0,q 4d, ... h, 
we have 
A =| |ay bg| [Oy co]... | | | + brady... %, 


[a bs | |dy eg| --- |r Js | 


cy OS Oy. Col an ech hi ta 
which establishes the proposition. 


49. Since by the preceding proposition any determinant of 
the nth order may be reduced to one of the (n—1)th order, we 
have another means of simplifying any given determinant. 
The proposition is especially advantageous in the reduction of 
determinants whose elements are given numbers. Thus: 


Asai 1 9}3.44 Aid —4 8) =— 4121 
3 Os iy4a apg ae et 1 <Sigeee 
1345 —6 —6 —6 1 ie 
043 2 

0 —3 


Here we can mentally reduce the determinants of the second 
order obtained by combining the first pair of elements of the 
first row with the corresponding elements of the other rows, 
and obtain the elements of the first column of the new deter- 
minant, thus: 1x2—3x2=—4;1x38—1~x 2 Ts eee 


—5x2=-—6. For the elements of the second column we 
have similarly: 2x1—2x8=—4; 2x4—83x38=—1; 
2x3—4x38=—6; and so on. 


Let the student apply the proposition to show that 
ey as | 1 1’ |=ayz;3 also/|}10 4°17 13) =ige 


Pete cel ce '9)1 4: 9 ee 
Poet sag 5. 81 8 <1 gee 
Liste eciagertit 2 7° 5 20 ae 


Also apply the proposition to show that 
510511 0 | = 2188, 


8-7> 2 20 
TP Da aa f 
841> O 6 


EXAMPLES. . 45 


MISCELLANEOUS EXAMPLES. 


XA * 1. Find the value of 


& |; -alsovof' |12-. 22) $4. .17-.20Raaa 

2 16—4° 7. .1 2h 
o9—l 6 2—1 5 10—38 —2 38 —2 8 

y (e125 8. 0 ae 


2 las 2 . 4° —8) ie 
Ble 2) x8 BAS 3689 OS 08 a 


maa 0 0 hetero, e000 ste ee 
Siew oo .. ee aay itihe 0; 02. 07.40 ,05RRO 
m1). 0 0 .. eats) 0. =O," 0, 2 ORO 
em —1 0 .. eres) Oe Oo. 0 ee 
eeu O.. O a, |, ja, 0° 0. 0...—d,_, bie 
MeO s..—1 - dm | - fd... 0 O .0 —b,, 


1000 aw~+thy +g9z|/=|100 awthy+gz+l 
0100 hx+ by + fz 010 he+by+ fe+m 
0010 gx+ fy +c 001 get fy+tcz+n 
0001 lu+my+ nz ey @ k 

oye 1 k 


4. Write the complementaries of the following minors of 


| Dy Co ds ey fz: | Ce G13 | Co Fs | 


Co Uy €y|3 |B, cg dg]; | de es fil. 


5. What are the complementaries of 


[Gyo Ass] and | dy ss Ase], in | Ap, Ayo Mog Agy Cys Ase | ? 


X~ Vv 6. Show that 
0 1 ] 1 Po p=2? 1+a 1 i 
1 0 l+a 1+0 1+¢e Tol -- bik 
1 a+l 0 a+b ate 1 1 1+¢e 
1 b+1 Ob+a 0 b+e 
11 e+1 ec+a c+ 0 


46 


THEORY OF DETERMINANTS. 


7. Prove that 


A, Ag Ag 
by by bg 
Cy Cg Cg 


dy dy ds dl, eee 


Lae 


dy bg| |r Bal fon Boies 


|. a, Cy] | a Cy 


‘Ge hI ie ls | ic ue 


8. Employing the notation 


show that 


i) 


_% (n—1)(n—2) ... 


; gate) re 
; ‘oe ear 


7! 


(Gets fae 


es 
Garg e c. 
ie 


wty+4 


e+2y+2 


(429 (=+9y oH 27 
: 5 


ztu—l a 
. = 


Observe that (" a ( 


The Product of Two Determinants. 


| Cy] +. 
| a; dy| [a ds| [a dy}. 


(n—7r +1) 


GOS AOI 


| (HD 
) 


ztu—l 
r—l1 


se 


| b,, | 


-[Q, | 
| d,, | 


ae l, | 


50. If we note a determinant by /r, and another by L, their 
product P is evidently expressed by 


ita ° 
Sims 


The form of this product suggests the probability that the 
product of two determinants may be expressed by writing the 
factors as complementary minors of a determinant of higher 
order, and filling in the vacant places due to one or hoth of 


GENERAL PROPERTIES OF DETERMINANTS. 47 


the factors with zeros. Suppose, for example, that A is of the 
third order, and Z of the second; then P would take the form: 


P= Ay b, Cy 
Ug by Cog Qa 
ds bs Cg 
Qa 
ROTA 


Qs Bs 


We now wish to discover if, when we fill in the vacant places 
due to KH or JZ with zeros, and thus make P a determinant of 
the fifth order, P will still be the product of K and LZ. That 
this is the fact will be shown in the next article. 


51. THrorem. — The product of two determinants, K and L, 
ef degree m and n, respectively, is a determinant P, of degree 
m+n, in which K and L are complementary minors, so situated 
that the principal diagonal of P is made up of the elements in 
order of the principal diagonals of Ix and L ; the vacant places 
in P, due to either K or L, are filled with zeros, and any mn 
Jinite elements occupy the remaining places. 


We have to show, for example, that 


Hox =| a, 0, 4, | X |. Bs | =| a0, G|0 0 O[=P. 
Ap bo Co Os Bs V5 Ag by Co 0 0 0 
ds bs Cg ag Be Ye Geb, Cg hr 00 
Cs by Cy) a4 Ba ¥4 
ds Ds Cs | a5 Bs Ys 
Ag De Cg | Ag Be Ye (1) 


Developing P in terms of the elements of the fourth column 
and their complementary minors, we have 


P=a,\a, b, 4 0 0 |—a ja, db, ¢,0 0 |+a,;a, b, ¢ 0 Of], 
5° 0a.Ca dew A, bo 40 0 Gg De-Cat Uae 
finde Cz 0-0 ds b, & 0 0 dy 0s C0) 
ds bs Cs Bs 5 My by Cy Bs 4 dy Dy Cy By ¥¢ 
de be Ce Bo Ye ds Dg Ce Bo Yo ds bs Cs Bs ys 


or Pe a; Aa, — 054, + a,A,,. (2) 


48 THEORY OF DETERMINANTS. 


But A.,= Bs| a 0 | 0 — Be 
Os De Loa 
A, bs Cs 0 
Ae De Ce Ye 


= Bs ye a by ¢3 | — Be ys| be Cz | = Kk 


ad, 0, 4.0 

Ay Oy Co 0 

As 0s Cs 0 

ds Os Cs Ys 
Bs V5 
Bs Ye 


In the same manner, we may show that 


A,, = K 


Bs V4 
Be Ye 


Bs V5 


Substituting these values in (2), we have 


P=K } 04| Bs y6|—5| Bs ¥6|+46| Bs ys|} = K Xx L, 


since the second factor is obviously Z, expanded in terms of 


the elements of the first column. 


The method of proof here given is perfectly general, and is 
applicable to determinants of any order. 
make y,=y;=0, and yz=1, P takes the form considered 


in 50. The student can readily make the application. 
As another exercise, the student may show that 


A=|a,0 q 
fe D16y 
Cn HOG. 
a, One, 


What difference would it make in the result if the zeros in 
the fifth, sixth, and seventh rows of A were replaced by any 


finite elements? 


$2. Writing the product of | a, 0, ¢; 


ance with 51, 


d,0O f, 0 
d, 0 fz 0 
ad, 0 f; 0 
dy. Ou F750 
0 e, O gs 
0 6 0 %& 
0e¢ 0G 


0 —1 
0 O0O— 
% Yi 
UY Yo 


= —| ay fo C3 Us| X | es Dg Gr 


Thus, if in (1) we 


” 


and | a Yo 2%3|, in accord- 


d 
4 
‘ 

“f 
a 
4 
; 
] 
; 
, 


GENERAL PROPERTIES OF DETERMINANTS. 49 


we have, by 37, 


eo 0 0 
0 0 
0 7 0 


AX + AgY, + 3% 0,4, + doy, + b,x 
Ay Ly + Ag Yo + M3% 0, Hy + Dg Ag + O52. 
Ch X3 Ug Y3 + g%z 0 Hs + doy + Dgzg 


0 —] (0) 20ge 
0 0 —1 0 
) 0 O-—1 


C1 Xy + CoYy + C324 My AYP ey 
C1 y+ Co Ya + Cys Uv, Yo & 
C1 Xz + CoY3 + C3%u BW Yn 


which, by 43, 


=| Oy % + Mg Y, + M32 01%, + O24, + 03% 6% + Co + 65% |. 
Ay Lp + Ay Yo + Age, 0, %2 + Do Yo + Dz%_ Cy Xo + CoYo + C32 
Ay @3 + Ug Y3 + Ag%3 0, Hs + Dos + D3%3 C1 %3 + CoYs + Cazz 


This result expresses the product of two determinants of the 
third order as a determinant of the same order. We are thus 
led to infer that the product of two determinants of any order 
may be expressed at once as a determinant of the same order. 

We now proceed to establish this important multiplication 
theorem. 


53. TuHrorem.— The product of two determinants, A, A! 
of the nth order is a determinant A" of the same order. <Any 
element a,, of A'' is obtained by multiplying each element of the 
rth row of A by the corresponding element of the .sth row of A’, 
and adding the products.* 


Before giving the general demonstration, it will be useful to 
establish the proposition for the product of two determinants 
of the third order, and note carefully the form of the result. 


* Forming the product by columns, the statement is, of course: The 
element in the 7th column and sth row of A” is obtained by multiplying 
each element in the rth column of A by the Corresponding element in the 
sth column of A’, and adding the products. 


50 THEORY OF DETERMINANTS. 
Put A= Ay by Cy and \! = ay By al ° 
a Dy Cy dy Pr» 2 
ds bs C3 as Ps Y3 


Applying the theorem, we have to show that 


+06, +471 (ly 09+; BoC yo Ayag+b, Bs+c, ys . 
Mga, +b2 3 y+ Coy Agag+by Bot Coys Cy 3+ Oo Bg + Cos 
3a, +b; Bi + C3 71 G3 2+ bz Bo+Cs Yo z3a3-+Ds B3+Cs Ys 


Since each element of A" is a trinomial, the determinant may 
be decomposed into twenty-seven determinants (35), the ele- 
ments of which will be monomials. But of these twenty-seven 
determinants only six do not vanish.* Those determinants 
which do not vanish are formed by taking for the first column 
a set of first terms from the first column of A", for the second 
column a set of second terms from the second column of A”, 
for the third column a set of third terms from the third column ; 
or, by taking a set of second terms from the first column of A”, 
a set of first terms from the second column of A", and a set of 
third terms from the third column of A"; and soon. That is 
to say, exactly as many non-vanishing determinants can be 
formed from A” as there are permutations of the numbers 1, 2, 
3, 7.e., 6. Hence 


AA'=A"= 


A" =| d,0, 0B, Cys|+| a1 Crye es 5,8, Aya, Cys 
Aza, O:Po Cors MgQ, CoV2 Y2Ps3 by By Agdg Coys 

Aga, O3P_ C375 | 3a, Csy2 bs Bs bsB, gag Cs'y3 

+ | 5,8; Ciy2 Mas; +} ay by Py dyag|+ CyY1 Aa, b, Bs 

bo By CoV2 Aga, CoV1 by By Ayaz CoV1 OUgag me 

bs By C32 Agas C371 0382 Mgag C3¥1 Ag ag bs Bs 


= a; Boyz (4 b2C3) — ay Bs 72 (Ay by C3) — ag Br yg (A102¢z) 


“i as By Ye (1 bo¢3) — a3 Boy (a, by¢3) + a 83 y1 (a by Cs) 
(by 30 and 31) 


* It is obvious that the determinants formed from sets of first terms 
taken from the three columns of A’, or those contai ing sets of first terms 
from two columns, etc., must vanish. Similarly for determinants formed * 
from sets of second terms, and 30 on. 


GENERAL PROPERTIES OF DETERMINANTS. ol 


= (a, bec) [a; Boys +a2 B37; +a; ; V9's5 a3 Beyi— a2 Bi y3—41 B32 | 
= (dC) (a; Boys) 5 


which establishes the proposition for the special case under 
consideration. 


In general, let 
A= Ay D, Cy eee l, and A! = ay By oval eee Ay 


Ap be Cy oes l, Ao Bo V2 ee Ag 
Cs bs Cg eee l, Ag Bs ¥3 eee As 
Ee Gn ys ee 6 a we iv 


Then the product AA! = A’ 
=] 034 +06, +G71+--. + Xa 


ga, + 2 By + Coy + ++ + lery 
Cgay + O38) + Cay + ++ + 13Ay 


Gy Oy -+ 6, Bi + Cnyit ois 5 At L,Ay 
A, + by Co + ye + eee + LX» eee a, + Bat C1Yat tha, ° 


Gin Oy + D2 Ba + Coo + +2. larg 026 Ayn De By + Co¥nt eet lrn 
Az Ag + bso ad C32 a eee + I.» eee Aga, + bs By + C3Yn+ weet leAx 


nO + On Be + Cayo «22 tly 9 22+ Ann +O, Bat Crynt «+e + brn 
Now, A" may be decomposed into a sum of n” determinants, 
the elements of which are monomials. But it is obvious that 
all those determinants whose columns are formed from sets of 
first terms of the columns of A’, or from sets of second terms, 
etc., will vanish, as each will contain identical columns. In 
fact, all those determinants into which A" is decomposed will 
vanish that have not the first column formed from a set of kth 
terms from the first column of A’, the second column formed 
from a set of rth terms from the second colnmn of A", the 
third column formed from a set of ¢th terms from the third 
column of A’, and so on. Now, as many such non-vanishing 
determinants can be formed as there are permutations of the 
_ numbers 1, 2,3... 7; that is, »! Hence, A” is decomposable 
into n! determinants, of which the following A, is the type: 


52 THEORY OF DETERMINANTS. 


Ay = b, By LA, 1 A3 eee Yn 
2[1 lgXy Agag «+» Con 
bs By l. No CAs Ag eee C3 Yn 
b,, 8, LAs An,A3 eee CrYn 
But Ar= By Agag eee Yn | Dy lo As eee Cy | 


Now, the determinant factor of A, is evidently A multiplied 
by the sign-factor (—1)?, in which p is the number of inter- 
changes of two columns which must be made in A to leave its 
‘columns in the order which they have in A;. Accordingly, 
Apr =(—1)? BiAgag...y,4. But (—1)?6,Agag...y, is a term of 
A', since the number of interchanges of two letters which must 
be made in a,Pyy3...A, to obtain the arrangement here given 
is p. Accordingly, A, equals a term of A’ multiplied by A. 
Thus each of the n! determinants into which A” has been 
decomposed is the product of A, andatermof A’. ».A"=A*'. 


Illustrations : 

j 1 20 3{x/|011 0\/=|0+2+4+0+0 3444043 
taeda Sag tl 0+14+1+0 38424140 
Boi. t | © Posey 0+0+2+0 9+404+2+4+1 
otal 2 rie 4 Wet RD) 0+0+1+0 0+0+41-42 


1+0+0+3 2+2+4+0+0 |=|2 10 4 4|/=4/1 
1+0+0+0 2+4+1+1+0 2 

3+0+0+4+1 6+0+2+0 212 
0+0+0+4+2 0+4+0+41-+0 1 3 


nD _ eR 
= CO 


(1000/|xk10100{=|0 1 1 1 : 
eA eG l10aa 1 @+a? ab+aB ac+ay 
Ob b B| 170" B 1 ab+aB 80+? be+ By 
ig 02 (Ce eae fy es Nee creed be fo ac+ay be+By @+y¥ 


34. Since, before multiplying two determinants together, we 
may change the form of one or both factors, the product of two 
determinants can be expressed in a variety of different forms. 
As an illustration, the student may verify the following equa- 
tions : 


GENERAL PROPERTIES OF DETERMINANTS. 53 


a 0, 
Ag by 


X | ay By 


Qs By 


=|)0,+6,8, dyay+ bi Bs 
Uy Q, + 6, By Cy Ag ate by Bo 
= 4 ay —- ty By (ly A + Ay By 
Dia, + by fon Dy eo) + by By 
=/da,+ ba, a6,+0,f; 
0+ Dya. dg By + b2B. 
— A101 + Agag a, By + As Bo 
| ja, + bya, 0b, 8, +b, By 


EXAMPLES. 


1. Show that one form of the product of 


fo bx 1 a*—a 1 |:is a? a—ab+b? a—ac+ec|. 
Lor 1 a’?—ab-+6? b? —be+c 
OT OT c—c 1 w—actc B—cb+e c? 

2. One form of the product of 
a+b e¢ ¢ |X|a+b+4e —ta —4b | is 

a b+c a —te b+c+tsa —tb 

b 6b c+a 


(a+b)? a Bf 


a (c+a)? 
-\ 3. Find the product of |a aaa|and|—1 1 O Oj, and 
i. abbob O—1 1 O 
al tre OP eT ey 
che Eee | Ce es 


thence show that the first determinant = a (b—a) (c—b)(d—c). 
4, Show that 


=e O° ¢.d!x Ll eee Led 
b—a ade e —1-1-1.1 
c d—a Ob —l- 1—i 1 
dc b—a —1 1-1-1 


=|{b+c+d—a a—b+c+d a+b—c+d a+tb+e—d 
b—at+d+e —b+a+d+e —b—a-—d+ce —b—a+d—e 
ct+td—a+b —c—d—a+b —c+d+a+b —c+d—a—b 
d+c+b—a —d—c+b—a —d+c—b—a —d+c+b+a 


| ie 


54 THEORY OF DETERMINANTS. 


= (b+c+d—a) (¢c+d+a—b) (d+a+b—c) (a+b+ce—d) 
1. las 


*) 


1 1-1-1 
1-1 1-1 
1-1-1 1 


and thence show that the first determinant 
= —(b+ce+d—«a) (ctd+a—b) (d+a+b—c)(a+b+ce—a). 
5. Show that 


i@isen 9a 20 Cl aot ore 
1 +R? —2b —28 +h? 1068 
1 e+, —2c —2y e+y Lica 
102+ —2d —26 baile ES Bay beg a Ae 
— 0 (a —b)?+(a— 8)? 
(a—b)?+(a—B)? 0 
(@—c)? (ay)? (=e)? +(B—Y 
(a—d)?+(a—38)? (b—d)*+(B—8)’ 


(a—e)?+(a—y)? (aa)? + (a—3)*| 
(b—c)?+(B—y)? (6—d)?+(B—8)? 
0 (o— 0) a 
. (c—d)? + (y—8)’ 
6. Show that 


Pete Cy Wa Xx pleO - 0. feet a ote 
Gg Da Cy 1 | WES o he 0-1 076 
ds Ds cs 1) 0.0. Lk, O50 sa8 
ise 10. PSH ST Bs | hy ig he A 


equals the determinant in Example 12, page 38. 
7. Find the two determinant factors of 
Cy Dye + CY, 01%+ Yo ay + C2- 0 fm+g%|. 


hy Do CoY DaQy+- Cos Uk, + bY. + Ce, AY, fX2+9%s 
As Dt + Csi) Vso + Cx¥/2 bys+cz, dys Gas 


; also of 


8. Form the product of | a 6,| and |a, ~; y:|- 
Uy dy ay Bo Ye 
ag Ps Ys 


The order of the first determinant may be raised to that of 
the second by writing it |a, 0, ¢|, and the product can then 

Ae Oo Ce 
Om Get 


GENERAL PROPERTIES OF DETERMINANTS. dd 
be found in the usual way. If we wish the product to contain 
only the elements found in the two factors, how should the first 
determinant be written? 

From this example it is evident that the product of any num- 
ber of determinants of different degrees can be expressed as a 
determinant of the nth degree, n being the highest degree among 
the factors. 


 ¥ 9. Employing the notation i=-/—1, show that the product of 


a+ib c+id| and | a—%tb, ¢—id, 
—c+id a—ib —¢,—td, aQtib, 
may be written DAs Bb Hae |, 
—b— 74. D+-iC 


in which 
A=be,—de+ad,—ad C=ab,—ab+cd,—ad 


B=ca,—qatbd,—bd D=aa,+ bb, + cc, + dd,; 
and thence show that the product of two sums, each of four 
squares, is itself the sum of four squares. (Kuler’s Theorem.) 
10. Show (1) that the product of | a,,c;| and |p, q.73| may 
be expressed as a determinant of the fourth order by writing 


the two factors | b, ¢ Oj and —| p, q% O 7,| respectively. 


Ge Oy Cy 9 Po Ge O Te 
fe bs C3 9 Ps Is 9 1's 
AS Et 8 Sa | Oo tO 
(2) By writing the two factors 
Gb; 6, 0 0} and 4p, 0-0 g, 7, 
dy 0, ¢C, \0. 0 pz 9 0 Qe Te 
as Dy &, 0 0 ps 0.0 Qs Ts 
Oy OS0e15.0 Ci Wee 0 
On On Dict O00 16:00 
show that the product is a determinant of the fifth order 


Gy OF Ge Ubue wand? 1 O°020-.0-..0 
ly De GU O50 Ome OOo 
dy b;-¢, 0 0:0 Otel Om 00 
0-0 OF Te0eG “000nR nN 
be-0; 50 -te > 0 0 O ps Qe Te 
OP OS0r OF 0 0 0 ps Qs 1°g 


06 THEORY OF DETERMINANTS. 


show that the product is a determinant of the sixth order. 
This example, and the theorems of 51 and 53, show that the 
product of two determinants of the nth order can be expressed as 
a determinant of each of the following orders: nth, (n+1)th, 
(n+2)th +» (2n—1)th, 2nth. 

55. Turorem. — Any determinant A may be expanded as a 
sum of products of pairs of minors. The first factor of each 
product is a minor of the rth degree, formed from a set of r 
chosen rows, and the other factor is the complementary minor of 
the first factor. The sign of a product is + or —, according as 
the product of the principal terms of the factors regarded as a 
term of A is +.0r —.* 

Every term of A contains 7 elements from the columns of a 
set of » columns found in the n columns and first 7 rows of A. 
That is to say, from every minor of the 7th degree formed from 
the first 7 rows, 7! partial terms of A can be formed. Now, 
the remaining (n—r) elements of every such partial term will 
be found in the remaining rows and columns after removing 
one of these minors of the rth degree. Or, in other words, 
(n—r)! partial terms of A corresponding to the r! other par- 


tial terms are found in every minor complementary to one of 
; 
the first set. Ce minors of the rth degree can be formed 
r!(n—r 
from the first 7 rows. Now, the product of two such comple- 
mentary minors gives 7!(n—7r)! terms of A; consequently, the 
sum of all the products gives n! terms, ¢.e., the full number of 


terms in A. 


To fix the sign of any product in this expansion, we have — 


only to remember that its sign must be the same as the sign of 
the product of the principal terms of the two minors. This 
latter product being a term of A, the sign of the product of the 
two minors must be the sign of the product of their pone 
terms, regarded as a term of A. 3 

If the selected rows are not the first s rows, we can easily 
make them so; then, after giving A the proper sign factor, the 
demonstration applies as given. 


* This expansion is known as Laplace’s Theorem. 


GENERAL PROPERTIES OF DETERMINANTS. 57 


Illustrations : 
Selecting the first two rows in |a, b,c; d,|, we have 
[Ay By Cs A] =| Del Les Eyl — | C21 Bs Ay] + 1a, do] |g C4 
+ | by eg! lag dy] — 10, del lag cg! +1 ce; do] |agd,}. 
Let the student select the first two columns of |a, d, ¢, dy], and 
expand, obtaining 
|@y Be! |g 4] — 1a Bs] | egg] + 1 ay Bgl | Co Ag] + | te Dg! c,d! 
— | a2 04| |e, ds! + 1a3 41 |e, del . 
Show that 
| abo, d4e5| = —|aod4| |e, d3e5| + lay cyl 10, d,e5| — |agdyl |b, c5e5| 
+ |a,e4| |b, ¢3d5| — | Boe! |ay des] + | body! lay c3e;| 
— |b: eql lay esd5] + ley dy! |, bs e5| + | cy €,| | aybsds5| 
—|d,e,| |a,b3c5!. 
What is the relation of 41 to the present theorem? 


56. It will be interesting to note what results, if, instead of 
multiplying the minors of the rth degree formed from 7 chosen 
lines by their complementaries, as in the last article, we mul- 
tiply every such minor by the complementary of a corresponding 
minor formed from r lines different from those first chosen. 
By the preceding article 

|ay ba egy es 

= | a, b4| |cgd4es| — | dy Bgl | eo yes! + 14 Oy! | Co d,e5| — | a, OI | cody ey 

+ | dy bg| |e, dye5| — | de d4l 10,3 e5| + 1205! 1c, ds eq! +13 04] |e, dyes! 

— |g ds |e, dyegl +1405! |e, dyes! 

Now, if in the above we write ¢ for b, it is evident that the 
determinant on the left vanishes, and hence the second mem- 
ber vanishes; but by this substitution we multiply the minors 
formed from the jirst and third columns of |a@,0,c;d,e;| by the 
complementaries of the corresponding minors formed from the 
jirst and second columns. It is obvious that the truth here 
exemplified holds in general. Moreover, it includes the special 
case of 45. 


58 THEORY OF DETERMINANTS. 


In symbols, the expansion of a determinant by 55 is expressed 
by writing DiS pa Nas tes 
where the chosen columns are two in number, or 

A= > | Ap b, Cy | Aa; bg Cry 

where the chosen columns are three in number; and so on. 

Employing the notation of double subscripts, we have, in 
general, 

A = | Ain | = > | Cp, G% Op, % pent? Opyay | Aap, q13 Apes 92> eee py dr+ 


57. THEOREM. — The product of a determinant A=|a,|, and 
any one of its minors M, of order m, is a determinant A' of 
order n+m. A's expressible as the sum of products of pairs 
of minors of A; the first factor of each product is a minor of A, 
formed from r chosen rows containing M, and the second factor 
is that minor of A containing the complementary of the first 
factor and the minor M. The sign of each product is determined 
as in 95. : 


Let the chosen rows referred to in the statement of the 
thecrem be the first 7; then, by 51, we have at once 
Al= 


OTy Og? +My yy, .++-Oyy-y Op) Oa 0, 0 
Ge} Ogg +++Ole, 1 Olgy +0eOgy Oey ang, e=s0a 0) ee 0 
Cy On,9 ee Up -1 Cy r+1 ee «Ayn 0 eee 0 0 
Uys11 An4r9¢ © +Upsy p-1 Onsy rie e*Upsin 0 eoe 0 0 
Uy, 1 Up_19+ ++ Up 1p] Up rye + © Up_1 p—1 Uy—ty Up ap pyee + Ay_iy 0... 0 0 
4 Ch, eee KJ nr e , 1 a eee Ann 0 eee 0 ®) 


oer re py dL, r+l1 
C411 bpp 190 eer p 1-1 Urp age ss Opty r1 Ur pie Uri npr +p in Urprie ++ bp 1 7-1 Urpin 


Ant Ung «++Anz1 Any o2eUny1 Any Anr+1 -+eOnn 
Oy) Spee Odes 3°* 50 QO | Oa 


nz +e*Unr-1 Any 
Uy se0Anr-1 Une 


Unsixes*Upsyr—1 Upesir 


Up _1y2 +0 Uy_y Pas eae 
Anz ee A, r—1 Uy 


GENERAL PROPERTIES OF DETERMINANTS. 59 


where the minor by which A is multiplied is enclosed. Further, 
observe that the n—7 rows of A not included in the chosen 
rows are prolonged in A’ with the elements of these same rows 
repeated in order of the columns beginning with the kth. Now- 
add the Ath row of A’ to the (n+1)th, the (k+1)th row to 
the (n+ 2)th, and so on, finally adding the rth row to the last. 
Afterward subtract the (n+ 1)th column of A! from the th, 
the (rn +2)th column from the (k+1)th, and so on, finally 
subtracting the last column from the 7th. Then 


A'= 
Wy Ao ++Ayp-r Un, ee Uy-1 Ur Ur + Ain Oh etree 0 ae 
NREaiDa ety) lo! 402g, Agy Ogy4] ---O,, OO . On Ged 


eee eee eos eee eee eee eee eee eee = hg is eee eee eee 


Cy Cn9 ee Cy hal Unn oe «Uy, r—l Upp Cy, r+1 eee Upn VU eee 0 0) 


Up Uy 9++ Ap k-1 Oran eee Apiay r—l Onsir Ansy r+1° ° Ansan 0 eee 0 0 


eee eee eee eee eee eee eee eee eee eee eee eee eee eee eee 


p11 Up—19¢ + + Up —1 p11 | U1 ye 6 «Up p—-1 Upp Uy 1rtys++Ay_in O .. 0 ) 
Cy] C9 ee -C,. zl Cx ee -C,. r—l Upp CL, r+l1 ee hn, 0 eee 0 0 


Oy41 1%,412°° «Uy 41 K-1 Does 0 0 W441 7-}-J°\ Artin Cy +iK 2 Opty r—1 U4 1, 


Anal Ong ee “An ia 0 e 0 0 An r+l1 ee “Ann nk ee On r—l Cina 
Pees, , OO. 0 O 6 gy ++ gy | eg - Ana Oe 
411 Up 419+ + Ans K-21 0 eee 0) 0 Ansir+ie+*Apsin As ine + Angi yr—1 Ansar 
Oy 41 Op] Qe AO Fs eel ) eee 0 0 Ay rtie° AL aly eo p_1Ke e +A,_1 r—1 Tp 
| hy. (1,9 oe oC; k-1 0 ee 0 0 C1, r+1 ee Ayn nn eee ‘y r—l ny 


’ By 55 A’ can be decomposed into products of pairs of minors, 
viz., the minors of the rth order formed from the first 7 rows 
and their complementaries. Since the elements in the columns 
of A’ directly below M are zeros, all the minors of the rth 
order, formed from the first 7 rows, will have complementaries. 
that vanish unless the said minors contain the given minor ©. 
I{ence the first factors of the products in the expansion of A! 
will all be minors of A, of the rth order, that contain the given 


60 THEORY OF DETERMINANTS. 


minor. Further, each complementary of such a factor is made 
up of the n —r rows of A not found in the first factor and the 
7 —k+1 rows in which M is found. Which proves _ 


theorem. 
Cy db, cy d,e, 0 0 
| ay Do Ge Oy nl 105 Gl = ly |B Cy| dz ee 0 0 
az |bs C3/ dz es 0 0 
by oe a @, by C, 


Cs b; EN hy 6s Die 5|— | Qy bs ¢| ld, €5 by Csl 
0 0 0 0 0 \Dy ol) —~ |x el eee 
0 05020 -9 [bs ¢ — |b) 5 eg | |@a GeO Gal 


| ay Bg C3 dy C51 | By Cg Ay] = |B, €g Ay | |e; Bz ez Fi; -+- |B, Coy &y| 105 Op eat 
The student may show (change the rows into columns before 
applying the theorem) | 
|, Dy €g Ay 5! | Dg Cg] = | ay Be ¢3 dy| 1s Cye51 — | ty Dg Cy U5! | Bg Cy Co] 
+ |a,b3¢,d5| |b,¢5e1 § 
| dy oC dy es | de 
= |¢, dyes! |a4d5 Ao] — | cy do 4! 1305 del + |e, d2e5| |d3 D4 de! 
+ |) ds e4| | dy D5 del — | Co dye5| 1a, b4 el + 1eodyes| 1a, b3dol, 
= —|d,6o| |dgb4¢5 del + | dyes! | b4e5 del — | dz ey| | b3¢5 del 
+ |d,es| |a,b3¢,dsl , 


The second illustration given is especially interesting as it 
shows the form of the product when the minor is of order n — 33 
In that case the chosen rows are n—1 in number, and the 
development consists only of two terms, each term being the 
product of two determinants of the (n—1)th order. If we 
change the order of rows and columns in the result, we have 

| dy By Cz Ay @5| | Do C374 | = | dy Do Cg Mg! | Bo Cy Ay 5] — | Dy Co Ms eg| | Ay Dg Cys! , 
or A Aa,, é; — Ae, Aa, _ Aa; Ae, . 
and, in general, 

A Aan; pq = Dax, Dang ma Naig Day 
Employing an obvious extension of the notation described in 
the latter part of 39, the last formula becomes 
_@A _ dA dA . dA dA 


da, da, Ang ~ ddy AAyg AA jy AA yy 


GENERAL PROPERTIES OF DETERMINANTS. 61 


Rectangular Arrays or Matrices. 


58. As a determinant is a function of n? quantities, the 
elements are always found in a square array. It is often 
necessary to consider the determinant obtained by applying the 
process of 53 to two rectangular arrays of elements, 7.e., arrays 
‘in which the number of rows is not equal to the number of 
columns. We will now investigate the value of this product. 

lst. When the number of columns exceeds the number of 
rows : 


The product of two arrays (matrices) of elements in which the 
number of columns (m) exceeds the number of rows (n), is a 
determinant which is equal to the sum of all the products in which 
the first factor is a determinant of the nth order formed from the 
first array (matrix), and the second factor is the corresponding 
determinant of the nth order formed from the other array 
(matrix). Let the two arrays of elements be 


yy Che aoe Cin eee Cin Oy, Ajo «+> Ayn ++ Ajm 
Gg] Ogg +++ Agn +++ Bom 


91 Ao eee Co», eee Com and 5 i} <i Mm. 


Oni Ane eee Onn =e nm J Onl Angers Ann sss Anm 


Applying the process of 53, we have the determinant 


A= 
yy yy Hee? FO Gin tes Fim 81m Oy, Aq + 28? + Ayn Gon 
Coy Ayo? + Aon Gin +2 ** + om 1m Cay Aq1 + *** + Aon Aon 


eee eee eee eee 


p11) -- am ac Cnn@1n et i nm 1m O19) ae gh ai Onn ®2n 


ft ++ + Om 2m ao. Ay Any tess +Oipy Onn *° tags Orin ° 
+ +++ + Com Com Pe Clg] Opi tee? + Aeon Onn + °** Fem Onm 


sae = Cnn Cor se Ayn On) + Hae are ot ke = fe nm nm | 


Now we may form from A a number of determinants Aj, A;, A;--- 
of the nth order, the elements of which are all polynomials con- 


62 THEORY OF DETERMINANTS. 


sisting of m terms each. The number of such determinants is, 


of course, Te eee = Basra ar: aS): Let us consider 
n! 
one of these determinants ; take, for example A,, whose columns 


are formed from the first » terms in the columns of A. We 
have, accordingly, 


Ay = | Ay ay tye O22 FAqy Oy 
Cbg 041 + Algg Ayg+ #** + Aon Ory 


On O11 a Ane 212 oe ane ar Onn1n 


lq] Aq3 F Ayo Gog 22°F Ayn Gon *** Ay Oni FOr Ano *** + Ayn Any 
Clg 9) + gg Ugg +** + Con Aon *** Cy] Any Ag: Ono *** + Aon Ann 


eee eee eee ee 


Oy 221 ots Ong A99 ee —- AnnGn *** Uni On = Ong Ong ne ce +n Ann 


Now A, is, by 53, the product of two factors, the first of which 
is the determinant formed from the first n columns of the first 
array of elements, and the second is the determinant formed 
from the corresponding n columns of the second array. Ina 
similar manner we may show that each of the determinants 
A,, Ay, A;++- is the product of two factors, each factor being a 
determinant formed from n corresponding columns of the two 
given arrays. Then in order to establish the proposition it 
remains to be shown that A=A,+A,+A,+---. Each of the 
determinants A,, A,, A;--- can be decomposed into n! non- 
vanishing determinants whose elements are monomials. <Ac- 
cordingly the sum A, + A, + A,+--- will contain 

m(m —1)(m — 2) +» (m—n+1) 
‘non-vanishing determinants whose elements are monomials. 
Returning to A, we see that it can obviously be decomposed 
into m” monomial element determinants; but those which do 
not vanish are only m(m—1)(m—2)---(m—n-+1) in number. 
Now observing that each one of these monomial element deter- 
minants is a part of that one in the series Aj, As, A3:++ in which 


its columns occur as parts of columns, the proposition is estab- 
lished. 


GENERAL PROPERTIES OF DETERMINANTS. 63 


Illustration : 


Performing the operation of 53 upon 


are f OMS 
we obtain the determinant 
Mo +08, + ey,+0,8, adyag+ 6; Bot yo + dy by 
Cyay + 2 By + Coy, + yd, gag + by Bo + C972 + dod, 
This determinant the student can readily show is equal to 
(@y by) (6 Po) + (A162) (ar y2) + (4 de) (a 82) 
+ (b,¢2) (Biy2) + (0142) (8182) + (¢1 de) (152) « 


2d. When the number of rows exceeds the number of columns. 


Consider the two arrays. 
Cy st ay P| 
(ty bg > and ag Bor: 
Ag bs Os Bs 
Multiplying as before, we have 


A= [dy +0) 8, Gag +b), dyag+ 0,83) = 0. 
Aly0 + 0,8, Azag+ beB2 das + 028; 
Aza, + 03R, Asa+ bsBy  dsaz + b3 Bs 
The value of A is readily seen to be zero when we notice that 
it can be obtained by multiplying two determinants formed from 
the two given arrays by prefixing a column of zeros to each. The 
method of proof employed in this special case is general. It is 
only necessary to add to each array as many columns of zeros as 
are necessary to make each array square, and then compare the 
product of the two determinants thus formed with the deter- 
minant formed by compounding the two matrices. 


Reciprocal Determinants.* 


59. If the principal minors of the elements of a determinant 
are themselves made the corresponding elements of another 


* Reciprocal determinants would more properly be considered in the 
next chapter since they are among the “special forms,” but for several 
reasons it is thought best to introduce them here. 


64 THEORY OF DETERMINANTS. 


determinant, the determinant thus formed is called the reciprocal 
or adjugate determinant. Or, in other words, the elements of 
the reciprocal determinant are the complementary minors of the 
corresponding elements in the original determinant. 


The reciprocal of (a, by ¢3) is 


(b2¢3) — (G2 ¢s) (Gz bs) |. 
— (0, ¢3) (a, €3) — (ad, b3) 
(b, C2) —(a 2) (a, bz) 


Assimilating the notation of 19, we have 
|.A, Bo C;...L4|, | Ain|, OF |Any Ass Aggy es eee ee 
for the determinant adjugate to 
| Gy By. Cy see 2, 1, | yn |, or | Gy Gog Ggy-ses) Cena 
respectively. 
If the minus signs in the first illustration are erased, what is 
the effect upon the determinant? How is it in general? 


60. THrorem. — The determinant A' adjugate to any deter- 
minant A of the nth degree, equals the (1 —1)th power of A. 


We have, for example, 


A =|a, 0, ¢|, and A'=|A, B, GQ]. 
Cig. 05) C5 A, By; Cs 
lgatat te Ag Bae GC, 
Whence 
AA'=|A 0 0/= A’. Pama a UE Wa 
0 AO 
OO” 


The process here exemplified is perfectly general, hence the 
proposition. 


61. THrorem.— Any minor of the kth degree of the reciprocal 
determinant A' is equal to the complementary of the corresponding 
minor in the original determinant A multiplied by the (k—1)th 
power of A. 

Let A =| ay], and A’= | A,,|. 


GENERAL PROPERTIES OF DETERMINANTS. 


65 


Transform A and A!’ so that the minors | dy dg 4, | and 
| Ay Ag Ay, | occupy the first three rows and columns in their 


respective determinants. ‘Then 
A=(—1)" | yy Ug Ay ys), 
|%s1 sg Asy gg 
Cg, gg Ugg hag 
Coy Ang Aggy Cog 
and A! = ( — 1 ) 4 hg As Av A; 3 
As, Asp As. Ass 
Ay Ay Ay Ag 
Ay Ay Ay Ags 
Then 
\An Asp Ay | cm = ie Ay, Ajp Ay Aj . 
As Aso As, Ass 
Ay Ay Ay, Ay 
) 0 0 1 
Multiplying, 
A | ‘Ay, Ans A,| =|A 0 0 Chg | = Cog a. 
Oe. Ase Ue ey 
Pa us is 
Vo Et Os © Wes 


Whence | Ay Ay Ags | = 2s A?, which is the required value of a 
first minor of A’. 
To find the value of a second minor of A’ we may proceed as 


follows : 
The minor 
| Ago Ass | i c— 1 Ay As An Ag, |, 

Ag Ag Ay Ags 
0 0 1 0 
0 0 0 1 

and the corresponding form of A is 

(—1)"|\ do. og ny hag | 

Qe G3 On Me 
ake igh: Cay 


66 THEORY OF DETERMINANTS. 


As before, 


A| Ase Ags| = A 0 Ao, Cog = A*\ay Chay 


O A Ge) ay 
0 O Gy Ay 


Whence, | Ag: Ags| = |G du| A. 
The student may put 
A = |a, by cs dy] and A'=|A, B, Cz D,\; 


and then show that 


| B, C; D,| =a 4’; 
| .A, ba | be C3 | A. 


The general theorem, of which the preceding are special cases, 


is proved as follows: 


Let A =|qa,,| and A’= |4,,), 


and let the minor of the kth order of A’ whose value is sought be 


*— 
A; —— Ap,o, Ap,qs Ap,4, ee Aya, _ 
Ap, Apa, Ap,q, ee Apa, 


Apa, Apa, Apa, cae A p,9,' 


Now putting 


p= Ppt Ps te Pues Vee Ue 


we may write 
ay — (—1)" 


Apia, s+* Upyax Up, +++ 
peg, +++ pede Apel ++ 


Cnn.g; eee Crd, Ap less 
ig, +++ Ug, CEL lunes 


Ap.q—-1 Apiqy+1 e+ Up,q—-1 Ap.qot1 
Up.q,—-1 Apoq,t1 os Cpoqo—1 Ap.qot1 


Cn .g,—1 Apz.qy+1 +++ Apyqo—-1 %pzq.+1 


C1q—1 Mgt +++ G1g,-1 A1g,+1 


(lag, +++ A2q, G21 «++ A2q,-1 Clgg,t1 +++ A2q,-1 29q,+1 


nq, eee Ong, Anil eee 


* In this determinant the subscripts p,, P., Pg, +» Jy» ox Yar «= 


eee Ap,n e 


eee Cyn 


eee Ap n 
eee Ain 
eee Con 


Ong,-1 Ang,+1 +++ Ang,—1 Ang.t1 +++ Ann 


stand for any integers in order of magnitude. 


of course 


GENERAL PROPERTIES OF DETERMINANTS. 67 


The corresponding form of A, is 


Ap,a, cae Ap,q, Ap,j ahd Ap,q,-1 Ap,g,+1 whe Ap,q1 Ap, qy+1 ree Ap,n ; 
Ap,«, siete Ap,q, Ap ee Ap,¢,—1 Ap,g,+1 isa Ap,q—1 Ap,.g.+1 ai Ap,n 
Ay, eee Ap, Ay eee Ay, 9-1 Ap, g,41 eee Ap qy-1 Ap, q.+1 eee Ap n 
eee -l oe.” 0 Orato. gr mnt) Cet sort 
tO. 0 Gagan sur rt Ue neea lt 
rene 0) ioc. 1 Line eee Oeigars oO 
een 0 1 ae 4) oe 
SIGS Se 0) ED Pees | Go erro 
ewer. ()... 0 omer.) Q er ye 
|, AS i @) Deeps. nO) OG ee vd Gon 
We notice that this form of A, is just the same as if it had 
been derived from A’ by making the p,th, p.th,---p,th rows of 
A’ the Ist, 2d, --- Ath rows, making the same changes in the places 
of the qth, qth, ... g,th columns, and then putting 1 for each 


remaining element of the principal diagonal, and 0 for every 
other element of the n —* rows of which A, is not a part. 


Multiplying, we have 
AA, 


a A 0 eee 0 Ap,1 eee p,q,—1 p.q,+1 eee Op ,qo—1 pn .do+1 eee Opin 


0 A eee 0 Clp.1 eee An.q,—1 Cp.qit1 eee Cp.qo—1 Op.qo+1 eee Anon 


0 0 eee A Cp 1 eee Cpy.q,—1 Cp.qy+1 eee Cpn.qo—1 pydo+1 eee Cyn 


= 
=) 
So 
Q 
— 
= 


. (1q,—1 Cl1q,+1 oe 1g.—1 Qig,tl ++: Ain 
so a C12] eee 2q,—1 29,41 see A2q.—1 2q.+1 ese Con 


0 0 eee 0 Anil eve Ang,—1 Ong,+1 oes Ong,—1 Ang +1 eee Onn 


of two determinate factors, and we have Y 
AA, = A* times the complementary of the minor of A corre: 
sponding to A, in A’. 


Whence seta ee 
A, = A*1 times the complementary of the minor of A corre- ee 
sponding to A, of A’, biel 


as was to be shown. a =% 


a 62. From the preceding article it follows at once that if ‘- 


ae A=0, then 
‘ Ay An = Ag Ag, | = ap 7k ae =0; 

Ay Age An) fa Be 

7.e.,-in general 
Ay, A,, — 

pk pe 

whence 
A; a Be = A, A,, ; 

or Ag ig ei as AL 


That is to say: Bit? 

If A=0, the cofactors of the elements ants any row are propor- , 
tional to the cofactors of the corresponding elements of any other 
TOW. : 


From the preceding article we have also 
Ag Ay 


; =A xX complementary minor of |dy, Ae ae 
; a Fes ‘pe ng Ane 
% which may be written 
3 do da|_, dA 
Ad, da, dada 
aa aa : 
day, Ady. F 0 ae 
whence ik CAs si dA GA Saas eric a on 4 


MMA, — Atyy day; AG, 
which is the formula already obtained in 57. 


GENERAL PROPERTIES OF DETERMINANTS. 


\ 


“1. Show that 
oe Oo 0 
Je 2, 
4 Ae 
1 he SaaS 
Zon Ue By 
Pave OU 
also 
te we 0 
Oe ay...) 
OMe 
foe) 0 
Ono. --0 
eur) 0, 
2. Show that 
C2 be 
Als bs 
aA, 0,4, 
es, 0B; 
HC, 0,0; 


Oro Se S'S 2 
~ ns 


Cy 
C3 


ob, B, B, 
CG 


EXAMPLES. 


0 0 
0 0 
seed, 


C, QO; 


69 


= — | Yo %s| | 20] 


=| a) b,| | cla ds| [as bel. 


= | A; Be C3] | dy Do C3]. 


3. If A is a determinant of the nth order, having n — m zero 
elements in the corresponding places of m rows, then A is the 
product of that minor whose elements are the other elements of 
the m rows and its complementary; the sign of the product is 


determined as in 55. 


4. If any determinant of the nth order has more than (n — m) 
zero elements in the corresponding places of m rows, the deter- 


minant vanishes. 


5. ty. bh 
My by C 
dz 0, Cg 
Gy Oo, 
ds 05 Cs 
Ch My Gs 
Bee. 0 
aU) 
d, d, 4d, 


M 
ie 
Cs 


N\=\|a,—M b, —N 
Q d—-P bh—-Q 


=|d, Os | [4G]. 


| Cz lg D5 | 3 


sy Ts: Gr Oa 
SR “A Slay a ee i te 3 

i * ~ 

“ 70 THEORY OF DETERMINANTS. 

‘x 

a 6. Gy Gg Ag Ug As Ag A Ag Ag Ajo| 

i by bg Dg 0g. Og. 206, by “2055.0 ae 
Oe OS 0 ie. ho Cs” 0 Cy 
Oo Duet0o O° 50+ 2.05.5~0y <1 el ee 
QO. AOE 0s OO es ra ep ee 
A Le Ss Se de Sele a 
f 92 9s 94 9s Yo Gr Gs O OY 
his he hs fg hgh hh, Sa eee 
bo ty. 30. 0 08 
ky ky 0-0 <0 (kh, ke 


= — |g dro] [Cod es] | fogs hs] |i *2}. 
7. Show that 


(a, — b,)? (ay — by)? (a, — bs)? (M4 a 4)" = 0. 
(a, — b,)? (a, — by)? (A, — bs)” (de = bx) 
(d3— 0)" (dg — by)? (dg— bg)? (ig — Ou) 
(a4 ree b,)? (a oe, by)? (a4 — bs)? (UM = by)? 
: This may be proved by multiplying the two arrays: 
; ; aR Cy 1 } 1 eed 4 db, oN, 
tt . ad ee Cp 1 an d 1 — 2 by be 
> * ae (ls iL . a 2 bs b.” - 
i Peed Mg | 1 — 2G a0s 
8. Show that. 
[ a, | (@ + Wy + ag + +++ + @,) 
SH] OyX] Ayo%y se Ain®_ |] Ay Ayg owe On 
Coy gg Coy, Cgy® Agee vee Agn®y 
Cnt Ang Grn Und Ong coe Onn 
ia Free +] Gy Ay Cn 
* Ca, Age Con 
2 1X AnoWo S AnnXy 
: Notice that the coefficient of x; in this sum is 
. y;Ay; + Ay;,Ao; + Os;Ag; + +» + Cig ling = | Qin |. 
. 9. As an application of the preceding, show that 
2(&, +22 +%3)|%, Wy My | 
a v3 @y 
aa r. 


GENERAL PROPERTIES OF DETERMINANTS. je 


Ry ay 2," 


—_ 
— 


oir oth wo Wz || @,W%y Woy Ly || % Wy Wy 

, . ‘a eo 2 

| ts aaa 1 a Oa ita ‘i 

10. Given Si(@) = a9? + 3 da? + 8 ae + dy, 
J2(X) = 093? + 3 ba? + 8 ea + dp, 
J3(&) = aga? + 3 b,x?+ 8 eye + da ; 
show that 
Ai(@) fil(x) fi"(x)|=—18[1 —a 2? —23}. 


Ta(a) “f'(a) Fy"(x) 
G2) fi@) Fa'(o) 


( Rae 1) Page ns 
The first determinant is at once reducible to 
—18 a2 + b, bette (etd, 


Ast +b, betec. a+d, 
dge+b, bse+te, c,e+d, 


’ 


which may be written 


1 0 0 0 ‘ 
mh G(e+b beta aqatd, 
Gp eth, dete cuet+d, 
dg At+b, baete cetd, 

Again using 37, the last determinant becomes the result above 
written. The student’s attention is called to the fact that the 
method of bordering a determinant, 7.e., increasing its degree 
without changing its value, here employed, is frequently of use 
in simplifying. 


63. The following examples comprise several interesting 
expansions of determinants. The cases considered and the 
methods employed are important. 


I. Expand the following determinant in ascending powers 


of x: 
A=lQ)t+t Ap .. Cin 
9} Ang +H ... Clon 


Ant Ong ein, Ong Toe 


oe THEORY OF DETERMINANTS. 


A is evidently a function of 2 of the nth degree, in which the 

coefficient of «" is 1, and the absolute term is f(0)=ja,,|. 

To complete the expansion, we have to find the coefficient of 2". 
Consider the product of two complementary minors of A, of 

the Ath and (n —k)th degrees respectively, 

Dee +e HO, and 

1 A Fy hy 


Oay Ugg +H ooo 


I eres vee 


Cog +O) Og, aaaes 


This product contains the term 


kt — nk 

| App Ang Un | = U'Dy_1y Say- 
| ek eats, 
Any Ong Cnn 


The entire coefficient of «* is accordingly =D,_,, 7.e., the 
sum of all the minors of |q@,,| of order n—k, whose principal 
diagonal lies in the principal diagonal of |q,}|. 


oe A= | Anl “< ®&Dr-1 ae x 3D,,-2 =e Ske + x". 
As an illustration, the student may show that 


Q, + & by Cy dy 
Qs GOs Ey es ds 
As Ds  G+a@ de 
Os Dg CG, (dee 


=|, bo¢3d4| + [| boesd4l + | ay Cy dl 
+ lay by Ayl + lay does |] x 
+ [ 1b. eg! + la, dy! + lay eg! 
H+ [bo dyl + lay dol + legal! ] a? 
+ [ay + by + c3 + dy]? + a4. 


For another exercise, let the student find the terms of 
A =!a,,| that contain # elements from the principal diagonal, by 
considering the product of two complementary minors, as above. 


GENERAL PROPERTIES OF DETERMINANTS. 73 
II. Expand 


A=|ax +ly Oe +my bw + my 
CUNY be +my awt hy 

Os + my adee+tiy ce + ny 

In ascending powers of w and 


ce first y=0, and then x=0, the terms involving 2 
and y’, respectively, are 


Pedecy 60; /.and-y° |b -ny om, |. 
Cree: a, Ho mG 
Gund. Cc Melo  % 


Putting the y’s in the two last columns of A equal to zero, 
we obtain for one set of terms involving 2 


wy|t oc |, 
. ealaig § Siaeat 
Me Oa C 


and the two other sets of terms containing 2’y are, similarly, 
wy 


a nm b,| and xyla oc mi. 
Co Mm ay Cy 04, 
eo... C by A, n 
The coefficient of xy? is found in a similar manner, and the 
entire expansion is accordingly 
A=~ala c, b,|+ay! | 2 ¢ d\+la n, b, +|a ¢ Mm) 
Co b ay Ny Ob ay Gg Ales CaP Crea den 
Dy A, C Mg. Ug C Dai ta a6 be As 1 
+ ay?! ja ny mi+l 2 ¢ mi+ n, 0,| 1 nm, ml. 
Geet thy ite Oe m a Ne me ty 
bo le n My Ay N Mts. le “ Me lo nN 


III. Show that any determinant A may be developed in terms 
of the elements of any row and column and the second minors 
of A corresponding to the product of these elements 

Let ' = |Ay1 Ag Ags] , 
and border it as indicated below ; 


calling the result A, we may 


[4 THEORY OF DETERMINANTS. 


expand A in terms of the bordering elements and first minors 
Bre. 2.e.55 
A =| Aor Bog Mog} = Moo A'— $ Gyo G1 An+ G1 Moe fas 
Ayo Ay Ayo Ng + Cy Qos Ayg+ Go) Ap, Ait M29 M02 Ay. 
Clon Ca, eg eg + Co Cog Ag+ Ago Mo Asi 30 Moe Asy 
Cso Clg, Ag2 Ogg + Glgy Cog Ags’; 
in which A, is, as usual, a first minor (with its proper sign) 
ofA7 
In general, if A’=! dy Gy++ G,, 1, we have 
va == Clog Coy Clog coe Clon = Cy) A! — 3 On ae ((,4=1, 2,3 eoeN)s 


| ly Cy, Cy eee Ayn 
Gig, Cay Onggy-0« On, 


eee eee eee e880 eee 


Ano Any Ang +++ Ann a 


’ 


in which, as before, -A;, is a minor of A’. 


For the terms of A containing @  are@bviouslyagA’. Now 


let C be the complementary minor of 


Os Og InAs 


Ai Ox 


then do, C contains all the terms of A involving da,,; hence 
a, C contains all the terms of A’ involving a,, and consequently 


C=A,, 


and — Gy o,A,, is the expression for the terms of A containing 
the bordering elements Gy, Mo, 

This expansion, known as Cauchy’s Theorem, is frequently 
written 

A= d,,A,,—20 doe (a) 

Here A is a determinant of the nth order. A,, is, as usual, the 
complementary minor of a@,, in A; ¢ has all integral values from 
1 ton, except 7; k has all integral values from 1 to n, except s; 
and (,,is the complementary minor of a, in A,. (a) is, 
accordingly, the expansion of A in terms of the elements of the 
rth row and the sth column. 


GENERAL PROPERTIES OF DETERMINANTS. Td 


The student may show that 


A=|a f g h|=abed — ffcd — gg,bd — hhybe. 
ho; 0 
Vom c . U 
‘Se FL Bae 9 ae 
. Brit ly 0 Ay dg: ~04| =| Oy-+ ty Oy. Og ai! 
aca ry Xo 0) o) ae vy Xo 0 0 
0 a te Xo Xe ©) ae Vy ) Xe 0 
@) ) — Xs U4 — vy 0 ®) U4 [ 


‘ Ge eda. ip re 
a a, E —+—+—+—>- 
1 Vy Vy Wy pe ah ng pe a 


IV ° If A — Vy Ag Og «+e Ay 5 
O41 Xo Ag eee An 
ay em) Xs eee An 
ay oe) Oe eee Xn * 


and if we put 


F(&) = (&@— 01) (@e— ag) ++ (Lr— On) 


and _ af(@) 
aA (2;) = a. = (&—a;) ++ (®;_1—a;_1) (Xi 410441) +9°(@,—an), ‘ 
we find A = f(%) + Sa,f'(%) . 
For 
A = | 1 0) ®) 0 ° ) = 1 aay —aQp, — Qs —Qy, 3 
i XY a9 Ag « Ay 1 X— Oy 0 0 
1 ay Xo ag « An Ni 0 vg— 9 0 0 
1 \ a, ag Xs +) 0g 1 0 0 L3— Ag 0 
fe, Oy ty de | ase) Be Taac0 0 0 sak Peg 


whence (if, as in III., we let A’ represent the complementary 
minor of the first element) A’= (wv), and, since every first 
minor of A’ vanishes except the minors of the diagonal elements, 
we have the required value of A on applying the theorem 


A = Ago Al— An Oy. Ags 


76 THEORY OF DETERMINANTS. 


V. Show that 


A=|a, 2% % @... 
Li tee Meee ae 
pee eg as 


= f(x) — af'(a), 


3 8 


i et ee a, 
in which f(x) =(#— a) (a - — ay) (® — ag) +++ (@— On), 
He) 


and f(a) = VO) = .) (@— 0g) = @ my) 
fen (& — az) +++ (@— an) 


deer (2 — a3) (@ — a3) +> (@ yg) 


A=/1 00 0... O|=|1 —x2 —2% = ie 
Lae fares oe lajy—az 0O 0 * cue 
ge rth ey yal 1 0 ‘@— 2) Oe 
Lee agers oat Le 0 -ag-— @ 27 0 


Lleawz..a| |1 0 0 Oj. 
Then, as in the preceding example, 
A = f(a) — af'(a). 
64. To the expansions of the preceding article we append the 
solutions of the following determinant equations. 
I. Solve the equation 


A=|x% a, a, a|=9. 
eG, ay 
Oyo: Gyo My 
a; a, a, @& 


We find by easy reductions 
A=(%—a,)*| & o a o|=(%—a,)3 (w+ 3.a,)=0. 
ad) | 


—t 
aL EO i Te 6 
Be hres Wht Ot 


Whence, C= Oy5 Byy Ay — 3 a}. 


GENERAL PROPERTIES OF DETERMINANTS. 17 


If. Find the values of x in the equation 
eae d, 6, ¢,(=0. 
mait. ¢, 0, 
ao, a 2 


a= e+ay+b,+¢, Ay Dd, Cy = (&@+ a+ b,+ ¢) 1 Ay by Cj 


L+ay+b+o « c dD, 1. ee Gees 
L+GA+n+¢q 4 © GY 1G) “eds 
e+aq+0,+¢q 0, a, & 1 0, a, x 
=(%€+a+0,+ ¢) (@—aq+d,—¢)|0 —1 1 —1). 
a er hr as 


EG ead 
Ee Oe dia. 2 
Put the two polynomial factors =A and B respectively ; then 
the last expression 
0 0 —1 0 
=A. B. 1 v os Cy Cy b+ Cy 
1 wt+eqg @ at+2 
1 Ob+a, a G|t+2 


0 0 —1 0 : 
(6 Gas eal 0 cq O&+Gq—a,—2 

1 0 x 0 

‘1lob+aq—-—%—-—q G 0 


Whence 
(@ + a+ O)+ C) (®—— +0) (+ %—-% — &) 
(q+e—b,—c¢)=0. 
6 X= — (t+), (h4—h4+), (4—-—G+Q), 
(b,—a,+¢). 
Ill. Find the roots of the equation 


A=| @ b° ee =} 
(a+r)® (b+ A)® — (e+A)? 
(2Qa+A)® (26+ A)? (2e+Ad)° 


From the third row of A subtract the first row multiplied by 8, 


78 THEORY OF DETERMINANTS. 


and from the second row subtract the first row. Then subtract 
the second row from the third, and we have 


(oi 


A=38» a 5° : 
B87+38acA+r 307°°?+30:4+H 38°+30aA4+)1, 
307+ ar 86? + br 3c? + cr 


Now subtract the third row from the second, and 


a? p° C = 0. 
2a+rA 26b+A 2t+aA 


BaP avr 380°+0A 38c?+ocr 


A=32% 


From this equation it is obvious that three values of A are zero ; 
the other two roots can be found by equating to zero the quad- 
ratic factor of the first number, and solving for X. 

A may, however, be further simplified as follows: subtract 
the first column from each of the other two; then 


A = 3d? (c—a) (b—a) a? P+ab+a e+ac+a?’ |. 
2a +A 2 2 


Sa?+ar 3b6+380a+X BeLraaeee 


Now subtract the second column from the third, and 


A=3A?(b—a) (c—a)(e—b)| V+ab+b? atb+el}. 
2a +2 2 0 
se°+ar. 8a+38b4+X 3 


Finally, add the second column multiplied by —a and the third 
multiplied by ab to the first, and afterward subtract the third 
multiplied by a+ from the second; then 

A=38)°(b—a) (e—a) (c—b)| abe —be—ca—ab a+b+c|=0. 
r 2 0 

0 r 3 


Whence three values of X are seen to be zero, and the other 
two roots are readily found from the quadratic 


(4+ b+ c)d*+ 3(be +ace+ab)rX4+ 6abe=0. 


GENERAL PROPERTIES OF DETERMINANTS. 79 


65. Tureorem. — The total differential of a determinant A 
is a sum of n determinants, each of which is obtained from A by 
substituting the differentials of the elements of a row for the 
elements themselves. 


Let A = | Yo%s +++ t, |. 
Developing in terms of the elements of the ith row, 
A= U%Xi+ YiVit %Z,++-4+ ¢,T, 
“. dA = dx, X,+ dy, Y,4+ dz;,Z,+ +» + dt, T,. 
There must be x h expressions for the total differential, each 
of which EGitionsly)s, after changing the elements of the 7th 
row into their differentials. 


*. [AA]* = | da, dy, dz, ...dt, |+| a y 4 ... & | 


olin Dea Se eae Me Uo, Geke san bp 
+ re ay! By: Ry ty 
Xo 2 &9 ty 


eco ieee je*8@ #88 ee28 


days dz... dt, 


From the differentials, partial or total, we, of course, pass 
to the corresponding derivatives in the usual way. 


Illustrations. 
MU. nq | M|. d|dM M\=|eM Mu. 
aa Sa —ldN WN dN N| |@N N 
Let | yy Ago Ag Ayl=A=\a 2b c Ol, 
Oras 2c 
eee ik: 0 
Uplie 20:.K 
nA: a 2b ¢ ibe c 
qqie at ae 56k 0 b k 
Gaede kh 


* The [ ] denote the total differential. 


80 THEORY OF DETERMINANTS. 


rik 2Ai+ 2g dn+Ag= 40|)° taal ie 


+2 bie | | rege 

es rs Meaty Ate el ar b k a 

. b Bek 
dA *¥, 
ae = Ay,+Ay+ 2 Aso + Ass = we 8g 
dA | oe 
—— —A pel CK ee om 
dk at An ; § 


66. THrorrem. — Jf the elements of A are all functions of the 


same variable x, = equals the sum of n determinants, each fhe 
ny 


which is obtained from A by substituting the derivatives of the 
elements of a row for the elements themselves. 


The truth of this proposition is evident from the preceding. 


Thus, if a 
AEE th) g(r) eons Tin OY te Se 


dS _| fa'(@) fra! (2) --Fil(@)| +) fal) ial) «+ Fin(@) |+ 
da Jat) Fa Bes » Fon) Ja () Fa (#) -. Sie @) 

Pec ave ee) fale) aon) ees 
+| fal) fio(@) «+ Fin(2) | 
Ja(#) Fala) «- + Fin) | 
Fral(@®) fal 2) ++ Sel @) a 
= AyAcds | 1 


Pree oe 
> 
(= 
So 


abil 
ee 
ci % 


If 


ee 
2 0. 0a 


as 


Bay cumecas' 
SGOHE- 


GENERAL PROPERTIES OF DETERMINANTS. 


the student may show that 


Peer) i- 2 ott « o| tein 
dx X2 ry ry 

iat ob cb Oita ee 

Xs As eae 


Gert vt Ch? Orage 0 QO 


81 


CHAPTER III. 
APPLICATIONS AND SPECIAL FORMS. 


67. We have now discussed the origin and some of the 
properties of determinants ; it remains to show how useful these 
functions are in application, and to examine some of the Special 
Forms that are of frequent occurrence. Within the limits of an 
elementary work like this it will be possible to select only a 
very few of the many important applications, and to touch 
somewhat briefly upon the special forms. Enough will be given, 
however, to enable the student to pursue his further investiga- 
tions with pleasure and profit. We now return to the problem 
with which we commenced the presentation of determinants, and 
proceed to the 


Solution of Linear Hquations, and Elimination. 


68. Consider the set of three simultaneous linear equations: 


Aye + by + CZ = My, Cy b, Cy . 
cathy teem met, and A=/a@, 0b, 6, 
Age - boy ae Coe = Ms As bs C3 


Multiply these equations by A,, A,, and A, respectively, and 
add by columns, obtaining: 
(Ay + d2A2+3A3)& + (b,A,+b,A,+b5A5)y 
~ + (A, +0,Ae +03.43)2 
= MA, + m,A,+™M;Az. 
By 45 the coefficients of y and z vanish; the coefficient of x 
is A =|a,0,¢s|, and the absolute term is |, },¢,I. 
Whence _ |m, by esl 
lay De Ce 


APPLICATIONS AND SPECIAL FORMS. 83 


If we had multiplied the given equations by B,, B,, B;, we 
should have caused the coefficients of x and z to disappear in 
the resulting equation, and would have found 


[ay Me Csi 


vc 
lay by Cel 


oJ 


Using C,, C,, C; as multipliers, we should find, similarly, 
_ lay by msl 
Go GeO cst 
69. To generalize the solution of the preceding article is now 


an easy step. Given 


AqyXy PF yoy F- +** Ay, Fe o0* + Ayn ®, = My, ) 
rs - es ie etl tp) PUPS AC le “ean = Mz 
° eee eee eee . ° : Ue F 
i oh ee Slop oe ay ORL ee ay each i 2 nae ee = ™M, 
AnyX a Peek ae seh Uy, ope lian we Ann Vy, = Mr a 


and A = yy yg eee yy eee Ain e 
Ag) o9 eee Qo, eee Con 


UO nhac GC, 


ny Ang ae One sii Onn 


Here A is, as before, the determinant formed from the n? co- 
efficients in the first members of equations I., and is called the 
determinant of the system. 

Multiplying equations I. in order by Aj,, Ay,, ... Any os» Anns 
and adding by columns, we find 


(QA), Se An Ao, a ee + AA, + Bc: -f- Oni Anr) vy 
a ee a5 eos +) + 2A. es nee Us 
ao eee vee 
or (rd i tos a ace a5 Avtaa nee a tal) 
ob eee Sas ofa 
=f ae ae ik 46 2 at ot Opn Lane + aes = (ate 
= m,A,, + mAo, +++: +m,A,, + + + m,A,,. pia) 


84 THEORY OF DETERMINANTS. 


In equation (A) the coefficient of all the unknowns except the 
coefficient of #, vanish, and the coefficient of #, is obviously A. 
The second member of (A) is evidently what A becomes when 
My5 No, ++. Mt, are put for the corresponding elements of the rth 
column. Hence 


X,. ae yy Cy eee My, eee Ain Cy Cy9 eee yy eee Cin ° 
91 Cog eee Mao eee Chon 9) Clog eee A>, ece Aon 
Cyg Cg see My coe Onn! ~ |Qpq Opg sos 
Ay Ane eee Mn eee CA, n On Ang eee nr eee Gan 


Translating this formula, we have: 


The value of each of n unknowns in a set of n linear simul- 
taneous equations is the quotient of two determinants ; the divisor 
(denominator) is the same for all the unknowns and ts the deter- 
minant A of the nth degree formed by writing the coefficients of 
the unknowns in order (i.e., the determinant of the system) ; the 
numerator of the value of any unknown as &, is obtained from A&A 
by substituting for the elements of its rth column the second mem- 
bers of the given equations in order.* 


70. The following modification of the solution already given 
of equations I. will be interesting. Employing the same notation 
as in 69, we have 


x, A — Ay) Cly9 eee (, x; eee jn 9 
Ao Ces eee Co, He eee Con 


Cy. Ogg *hss , Opptiere memes 
: Any Ang eee Oe x, eee Lg Re, 
which, by 37, 


* This is the rule for the solution of simultaneous linear equations first 
obtained by Leibnitz, and subsequently rediscovered by Cramer. (See 
opening paragraph of Chapter I.) 


APPLICATIONS AND SPECIAL FORMS. 85 


F/O Aya vee Ayp-y Ay Ly Ayo Woes Ay Vp Ap Bp 
Gay ag e+e Aap Ag, Vy gg LaF +++  Ayp_y Vp_y Hb Ag Vp fo 


1 C9 coe 4 San 1 vy a C9 Lot eee a Cnt HP + Close x, lL ase 
Un Ang os Any—y Any Vy + Ang Xe fee +r Uny—y Vp] soe Any W, es 
cE iy Sa iecimarey ted ¢ Ppl 


“f- Aon Xn Cgr44 zone Con 


+f Ayn Vp Upp) coe Ayn 


=f Ann Vn Cnr+1 coe Onn 
Now substitute in the last determinant the values of the 
elements of the 7th column, and 


a, A = yy Cp eee Ujy—-1 My, eee Cyn 
mecartess Coro Mla. .: >" Cox 


9 


te, thee Ceca) AN sss. Oy, 


ny Ang e+e Anp_y My «ee Ann 


as before. 
A simple example of the methods of 69 and 70 is the solution 
of the following equations : 


5X+3y+3z= 48 5 3 3| 
2%+6y—8z2=18?7?. HereA=/2 6—8)=—231, 
Petey oe 21) hed 2| 
48 3 8 5 48 38 5 8 48 
18 6-38 2 18—8 226.513 
pete 2} egresle Boe i wi 8-3. 21 
oe  — 2381 —231 —231 
As another example, we may solve the equations : 
y+2 fonb | bed A ee | 
2tut«e=b p@ 1001s Lee 
ees Here A= 1-0. Be 3. 
e+y +2 are, Pe eg 0) 


* 


86 THEORY OF DETERMINANTS. 


The student may show that 
e=4t(b+ce+d—2a); y=st(c+d+a—25); 
z=4¢(d+a+b0—2c); uwu=F(a+b+c—2d). 

71. We have hitherto tacitly assumed that neither A nor m, 
({=1,2,...) should vanish. If A vanishes and m, does not, 
the value of each unknown becomes infinite. If m; vanishes. 
while A does not, the values of the unknowns are severally zero ; 
but when m, vanishes, the system consists of homogeneous 
equations, and their solution is given later. If m,; does not van- 
ish, but A and the numerators of the unknowns do vanish, then 
we have the following theorem. 


72. If the equations of a set are not independent, i.e., if any 
one (or more) is a consequence of the others, the value of each 


0 
unknown takes the form —- 


Since the equations are all linear, any one can be derived from 
the others only by the addition of two or more of them after 
each has been multiplied by some constant factor. But this 
gives rise in the determinant numerator and denominator of the 
value of any unknown to two or more identical rows, and hence 
numerator and denominator vanish. 

For an example, take 


Ay X +:0,;%., +O, % =m, A 0, Cy 
ly % + Oo%, +65% = Me ; where A = | dz by C|= 0. 
42a + Ad, Bot A, CO, Ve= A, My a, b, 
We find 
M, 6; 
i Deets 
i ee =9; m= oe wt, = 102 Oa Mal _ 0. 
A A 0 


For a second example, the student may show that the values. 
of the unknowns in the following equations take the form >. 
38x+2y—52e= 4 
6x—8y+4z= 22}. 
Y 22=—2 


APPLICATIONS AND SPECIAL FORMS. 87 


73. If m=m,=:+--=m,_,;=0, and one mas m, does not, 
we evidently get 


dl ea MyAny apc MryAns 5 Sak MryAnn 
Beets AT? ee Gray Oy ee 
v x. x m 
ea St I A ie 


Ay A n2 Fh A 


74. If m=m=-+-=m,=0, te, if equations I. become 
homogeneous, then, unless 2, Xo, +++ x, are severally zero, A must 
vanish, 
In that case, equations I. become 
Ly S Oy Wy yy Wo +++ $F Ay, oe + Ayn Ly = 0 
Ly = gy Ag Lot +++ + Any Lp *** + gn Ln = O 
eee eee eee . . eee eee eee eee Il. 
DS yy By yy ob +0* Fy, BP ee Lb Ayn Vn = O 
is py + An9X_+ Se oy Qapte** 4 CnnLy, = 0 
Since a, A =| dy; Algo gg +++ M,*** Ann»| =O (m, being zero), the 
truth of the assertion is obvious. 
An example is furnished by the homogeneous equations : 


My L + Ay % + Hyg %3 = 0 ) : 
C9) Ly + Clo Xo + Clos Xo, = 0) ° (£) 
151 ® + Cg9%q + Ass Vs = 0 


Multiplying equations (HZ) by Ay, An, Asi, respectively, and 
adding by columns, we have 


(ayAyy + yA + 3; Agy ) @y 
+ (ygAy + Mg9 clo + Ago-Ag1) Xe 
+ (dig Ay) + G23) + A33451) v5 = 0. 


The coefficients of x, and w, are zero, and we have 
ae i Cy Ae 
As a further illustration, the student may show that if 
nae vyy +w22 Uy (YaAtypz) HU (2a, +22) +W, (LJ, +MY) 
is zero for all values of x, y, and z, then 


wow — Uuyz— vo — wwPt 2 uUvwW,= 0. 


agit 


88 THEORY OF DETERMINANTS. 


Observe that by the given conditions the coefficients.of 2, 
y, 2 voust severally vanish. 


75. With the help of 74 we obtain an interesting proof ot 
the multiplication theorem of 53. Consider the simultaneous 
equations 

(4 —A)%+ - 0% + 4% =0 | 
Ay%, +(bo—A)Xe+ Cot, =O Pl. 
3% + b3% +(C3—A)X= 0 


By the preceding article we must have 


A= a;—X by Cy 9 OF 
Cy bo—-A Ce 
Cs b; C3—X 
or W—-MN’+NA—-P=0; (a) 


where we notice especially that 
P=|a, by ¢3\. 
Let the roots of (a) be Aj, As, As; then, evidently, 
: P= — Qi dg Aj: 


Now, from I. we obtain three new equations as follows: 
Multiply equations I. by a, a, a3 respectively, and add them 
together; also multiply equations I. by f;, Bo, Bs respectively, 
and add; finally, multiply equations I. by Yi» Yo. Ys respec- 
tively, and add. We now have three new equations where the 
determinant of the system is 


A! = | da, + day + Aza3 — a, Ar by 0; + bay + bag — aor 
A By + A Bg+ a3 B3— BA 0, B+ b,R.+ b383— Bod 
Yi + A2y¥2 + Asy3—yirA by, + Dove + bs y3 — yor 
C101 + Cody + C303 — a3r|= 0, 
C1 Bi + C2 Bs + C3 Bz — Bsr 
Cry + Coe + C373 — Y3r 


or QM’ — M2 + NiA— P,=0, (0) 


APPLICATIONS AND SPECIAL FORMS. 89 


where we observe that P, is what A’ becomes when we put 
n= 0), and that 

Q=la, Bo ysl. 
Further, since 


Lip a Ag Agathe, 


it follows that 
P,= PQ=1m% be 31 Xla, Bo 31. 


But P, is exactly the determinant obtained by 53, and this 
was to be shown. 


76. The condition A=0O being fulfilled, the equations no 
longer determine the actual values of the unknowns ; they deter- 
mine only the ratios of these values. For, if 2,', x,', ... 2, 
satisfy equations II.,so will ka,’, ka,',... kx,', k being any factor. 
Any n—1 of the given equations will suffice in general to de- 
termine the ratios of n —1 of the unknowns to the remaining 
one. An example will make this clear. We employ for brevity 
only three equations : 


Ay ® + doy + C2 = 0 
a3x + bsy + 62 = 0 


Write these equations 


het bhy+oq2=0 
(a). 


x 
ee ae Spe 
dy ~ + = — by (0) 
Y y 
wv “4 
dg—-+e,-=—b 
ay | By 3 


From any two of equations (b) we may find the values of 


©, ~; thus from the first two 
y y ‘ae 


a ae ast s 
a oa ee 


: th anne? Sis Pe rk ee eee Nee 
; - i; Teas hg tee 
“3 2 a ee 
_* 
90 THEORY OF DETERMINANTS. 


Again, en (b) are to be simultaneous ; ; hence these 


values of = and = must satisfy the third equation. 
y Y | 


Substituting, 
Cs | Dy Co| — bs| dy Co| 4. C3| Ay by| = 0 


or A=0. 


Since from the preceding equations ; also equals ° 


|b, cal lb, eal 
Poorer 


we have 


In the same way, 


Be Oy Gs 

y B Bb B 
and hence 

@_A,_A,_ Ay, 

4 1 C, Cr 
or, 


Oss wen Abe Cs 


That is to say, The ratio of any two unknowns in a set of s 
homogeneous equations is equal to the ratio of the cofactors in A 
of the coefficients of these unknowns in any of the given equations. 


The general proof of the proposition just stated may be given 
: as follows. We have to show (equations II.) that : 


a 
Wy? Wg Ugirert Mt eet My—= Ay? Ajo: Ajgi s+: Ayes est Ay 
— e . . . . e Lay ¢ 
ae a As: re As, : eee 5 Ass a 


rie Die: Aas . «+ eee +t Ayn a 
If these proportions are true, we must have the equations é ss 


©,= XA, (A = constant; p=1, 2,-+- 7). 


mo 
‘he 
CT 


APPLICATIONS AND SPECIAL FORMS. 91 


The equation 
AyAyy + Ay2A ye + +++ fb ppAyp + 2+ + Ayn = 0 (Ee) 
is always true, whatever the value of p, since A is itself zero. 
Substituting in (#,) the values of 


eas A.;, ee A 


“pry 


as obtained from (£,), and multiplying by X, there results 
AyHy yy A hy gly 29% Ay, oe + Ann ®, = 0. 


This last being a true equation, the proportions from which it is 
derived must hold.* 


77. From the last article, or the two preceding articles, 
we deduce the important conclusion. Jn order that n linear 
homogeneous equations may be simultaneous, tt is necessary and 
sufficient that the determinant of the system vanishes. In that 
case any one of the equations is expressible linearly in terms of 
all the others, provided the first minors A, do not all vanish. 
For we have in general, A being zero, and |, J, ...l, repre- 
senting the linear functions of equations II.,_ 


LA, se 1, Any of pa 27 bithons = 0 ; 


hence, if one at least of the first minors A,,, Ay, ...A,,is not zero, 
as for example A,,, /, must be expressible linearly in terms of 
Pele, --.t,, and, hence /;=0 is superfluous. If all the first 
minors vanish, and one at least of the second minors does not, 
then, similarly, it may be shown that two equations are super- 
fluous, the system being doubly indeterminate, and so on. 


78. Among the proportions of article 76 consider the 


following : 
Hy 2 Vy: Wes res Ly =A, ; A,» : Ang are leas CB} 


* This demonstration applies of course so long as the first minors of A 
do not all vanish. 


4 >” v 
co 


: 


92 THEORY OF DETERMINANTS. 


Ayn, Anzy Ang; *** An, are, none of them, functions of the 
coefficients of the last equation of set Il. in 74, 


Any Ss OngX2 = Ong Xz a se + OnnXn = 0. 


Hence, proportions (P) give the ratios of the unknowns 
1, Wo, Us, +++ &,, that satisfy the n—1 equations 


4% + AyoXo + ae be Ay,t, = 0 
Ay, + AgV. s+ + Ap,X, =0 IIl., 


eee ee eee eee 


On 11% + Ay jotg-+ ‘eS se An —1nXn = 0 


if we denote by A,, the determinant formed from the co- 
efficients in equations III. after suppressing the first column of 
terms, by A,. the determinant formed from the coefficients of 
equations III. after suppressing the second column of terms, 
and soon. Hence having given n homogeneous equations con- 
taining n +1 unknowns 


yy Uy + Cj9 Xo + tS + Ajn+1 Unt = 0 } 
Ag, Hy a C9 Xo ~ gee + Moni Unit — 0 IV; 


Any XH + Ang Xo ae oe =e Onn+1 Cnty — 0 
we find the ratios of the unknowns as follows: ¥ 


ae im] 
put Ay= (1) 741 Gy Ga, ea Aniny °° Canty |. 


Gg, Mog 22% Gg;_1 Cais, *** Cgniy 


ny Ung *** Ani nist wet Cnn+t 
Then from what precedes 
Hi Hy givers Wyiy= Ay: Ay: Agi +e: Anis. 


79. Consider the following n equations containing n—1 
unknowns. 
MX Aye. bes + Oy r%i1 +P =0 
Api AgpXy + +++ + A, 1%, +p, =0 
Ay —11% “F An yo g + +++ + An_in 1% n_1 + Pn1= 0 | . 
AnyXy Ano. tere + Ayn 1X1 +p, =0 


’ 


APPLICATIONS AND SPECIAL FORMS. 93 


Equations V. may be made homogeneous by multiplying them 
by uw, and regarding a,u, xu, ...%,uU, wu, as the unknowns, w 
being any arbitrary quantity. Whence, if these equations are 
simultaneous, we have by 77 


Os a Pits je 0: 


An—11 An—19¢*+ A@n-in-1  Pn—1 
Ca CRANE aly | 

This result may be expressed as follows: n equations (not 
homogeneous) containing n —1 unknowns are simultaneous if the 
determinant of the nth degree formed from all the coefficients 
(the second members of the equations being included among these 
coefficients) vanishes. 

This condition could also be derived from equations IT., Art. 
74, by putting 7,=1. Those equations, m in number, then 
contain n—1 unknowns; and if the equations are simultaneous, 
we see that |q@,,,| must vanish. 


80. With the help of the preceding article another solution 
of a set of linear equations may be obtained. For brevity we 
employ only three equations : 

(1) A,X, -b D, X» + C, Vs — My 
(2) Gy% + by Aq + CoX3 = Mg 
(3) G32, + b3X2 + C3%y = Mg 
Take with these equations another, 
(4) gay + Dy Wy + Cy M3 = M4, 
which we suppose consistent with the first three, and in which 
Gy, Dy, Cy, Ms are undetermined. By 79 


Oe 6. -t.t = 0; 


or, tty Ay + b, By +oC, + m,M, = 0; (5) 
where, as usual, 

A,=—|b, & ms} 3) By = lah Cy Mel 3 

O,=—I|a, b, Ms! ; M,=\aq, by C3) = A. 


94 THEORY OF DETERMINANTS. > 


Now if we eliminate m, from equations (4) and (4), we get ea 


a(ats) + (t+ 3 a) ta(at 3)= 0. 


(6) a 

Since equation (6) must be true whatever the values of ay, b,, Cay a 
the coefficients of a4, by, c, severally vanish. cP 
Ay, By, 


. Seria trace a Saas BH 
__ lay Me Col __ |@; by mel 


|, Dy Cs] , 
me PEG TL R , 3 iN 


or, 
A 


Wy = 
e 81. Let us now return to equations I. Art. 69. Considering © 
M1, Mo, Ms, ... M, aS linear functions of the 2’s, we can express 9 
any new linear function 


% C1 Ly Co Hy + 22+ +6, 2%, = Y 


in terms of the m’s. 
Thus, if we have given 


+ Aint, = ™M, 
+ Ag, Xn = Mo 


C0, + CMe 
Ay LX} + Ayo%g + +++ 
Cbg Ly Aggy + 


ao 


Gini X + AngXe+ °° 


+ Ann Xn = My, 
by 79, 

A! Cr Y 
Ay, My, 
Clon Mg 


Cnn My 


Now if A=la,,|, we readily obtain 


Ait yA=+ yd; 
or, + ya st Cy Co eee (bs, 0 a 
yy Ayo eee Ahn My, 
A) Cg9 eee on Mg 
Any id “we On My 
*; rice s, | Pit teeta eae ee 


APPLICATIONS AND SPECIAL FORMS. 95 


82. We have seen that if n homogeneous equations are to be 
eonsistent with each other (simultaneous), the determinant of 
the system must vanish. The equation > 


A=0 


then is an equation of relation between the coefficients, and is 
really the result of eliminating the unknowns from the given 
equations. We shall soon investigate this resulting equation of 
condition or resultant in detail. We here deduce a general 
form by which the result of eliminating n unknowns from p 
given linear equations, supposed simultaneous, may be ex- 
pressed, p being greater than n. 
Given 
yy + Ayy%y + +++ + Ay, %, = 0 
gy Hy + Ang Ly + +++ + Ann X, = 0 
a sie : La VI. 
ny ® K Ang tg bees + Ayn %, =O 
hy X + Aga Xe 206 + Ayn, = 0 


If these equations are to be satisfied for other than zero val- 
ues Of the variables, the determinant of the system for any n 
of them must vanish by 77. The equation expressing this con- 
dition is obtained by writing 


Cie igen ses Oye 1 =. 05 
C9) ery) eee Clon 


OL OM gees ad 


Equation (J) is accordingly interpreted to mean that every 
determinant of the nth order formed from any n rows of the 
matrix on the left must vanish. For an example the student 
may eliminate the two ratios 7: v,: x; from the five equations 


ty + 0; 05 42 G2 = Oo (il, 250. )s 


obtaining the equation 


Pee 
G db, ce |=, or |) a, Gg Ay My Gs || = 


lp Dy Cy b,- 0, 0; “Oy. Been 
OM Of % 
ds; 05 Cs 
| a : a ib 
f 83. Suppose we have given Boer! 
4X + Diwe + CX ole dW, = 0, A 
AgQy + Doe + Cos + Ape, = 0, akg 


Ag, + DgQy + Cy3 + Asx, = 0 ; 
then, by 78, 


Wy Wy: Wy: y= 1D, Co dsl : —]y Cy dsl : 1d, by dg] : —lay by Cyl 


Substituting the values of 


we get the relations 
dy | By Cy ds3| — by | Gy Co dg] +] a, b. ds] —d, la, by c,| =0, 


Gz | Dy Cy ds | — by | ay Cy dg | + C2! ay by ds| —d,| ay, by Cg is 
ds | by Co ds | — b3| ay Cyd, | + C3] dy Os | = alae bay] =O ; 


which are all expressed by the matrix 


To generalize this, we return to art. 78. 
From equations IV. we found 


U3 Xo: Hi eee Vn = Oy? pat A, Lei Ans 


Substituting in equations IV. the values of 


oh mee Cn +1 ed Se ae 
—_— —_— eee 3 Pe ee aa 


BP sciee 2, 
v, wv, vy, iw $ x ‘ 


APPLICATIONS AND SPECIAL FORMS. 


iwi 
if 


2 ven by these proportions, we have 


My Ay + MypAg + e+ +A, + oe) + Ging Ang, = 0 
yyy + ggg + +++ + Aly, A, eves + Gani Any = 0 


Ay 9 yoo ae Ne =f Dry Ay “ie ae A =f Onn Mngt =0 (a 


By yAy + Ando + +++ + yA, eee + OnntiAnt = 9 


These n relations are expressed by the matrix 


Oy, Ayg 29° Ay 29° Ay Anyi = M. 


Qon+1 


Ayn +1 


eee eee eee eee eee eee eee 


ny Ong °** OAnp *** Onn Ann+1 


7 _ 


- 


_ We have accordingly, in general, from a matrix of the form 


; 


M, the following relations : 


| Che Ay see Cy, eee Ain Anti — 9 yy yg coe Cy, eee Ayn Oyn+1 
| Gag Clgg +2 Agr °° Gon Clon+1 |g, Meg *** Ay *** Aon Aon41 


1Ayg 3 eee Cy eoe Onn Ont 1 C3 eee Orn eee Aen Opn 
Ang Ong eee np eee (8 eee Onn+1 Ani Ong eee np eee aw Onn+1 


+— eee + (—1)" Gna yy Ay a J Cy, ie hs Cin =, 
Gg, gg *** Any = 2** Con, 
Uy App 28% yp 28% Ann 


Oy Ong *** Unpr °°* Onn 


ich v has successively all values 


. 7 
. 


98 THEORY OF DETERMINANTS. 


84. We will now select a few examples to illustrate the 
foregoing processes from the vast field of application. 


I. To find the condition that three right lines shall pass 


through the same point. ne ee 
Let Me +by+ec¢=0 
cathy t= | (A) 
ase + by + Cs = 0 


be the equations of the lines in cartesian co-ordinates, and let 
2, y; be the given point. Equations (4) must be satisfied for 
C= 2%, Y=Y,; hence 
ae, + dy, + ¢ = 0 
Cg + Do, + Co = 0 
Ast, + Oy, + C3 = O 


But in that case, by 79, 


(B) 


| b.¢s| = 0 
which expresses the required condition. 


II. To find the condition that three points shall lie on the 
same right line. 


Let (%, Yi) 5 (2p, Yo) 5 (23, Ys) 


be the given points, and 


wine are 


ae + by +¢,=0 


the equation of the line. Then 


a0, + by, +c =0, 
Ugly + Deo + Co = 0, | 
Cbs3 + OgY3 + Cs = 0. 


Whence the required condition is 


Oe gies) aay 
® Yo 1 (R) 
% Ys; 1 ( 


APPLICATIONS AND SPECIAL FORMS. 99 


As an application of the present example, we show that the 
middle points of the three diagonals of a complete quadrilateral 
lie on the same straight line. 


The three diagonals being OC, BA, B,A,, and their middle 
points #7, D, E, we have to show that F, D, E are on the same 
right line. 

Take the vertex O as origin, and the sides OA,, OB, as axes 
of reference. 


Put a, =OA4, ea Be bes OD, = OD, 
The co-ordinates of D are re 2, and the co-ordinates of E 
are “2 The abscissa of F is half the abscissa of C, and 


the ordinate of F is half the ordinate of C. Hence we have 
to find the co-ordinates of C. The equations of AB, and 4, B 
are respectively 


x 47 = 1, or O0¢-+ay=a,0,; 
a 0, 
behets 1, or bdye+ ay = ab. 
dM Oy 


Whence the co-ordinates of C are 


(D5 Cy bo (yD. 
aD -Gs Db, eb, 
= eee yf 
by = My lb, Gy 
Bit yds b, ae 


100. THEORY OF DETERMINANTS. 


and the co-ordinates of F’ are 


aay Ay (b, ss by) G; (dy ee a) 
2 (bsg facad bid) 2 (bo0.— bya) 


Now, by equation (#) above, 


A= ay by 1 =, 
2 2 
Og 2, 
2 5 : 


Ay Ay (by — ,) Db, bo( dz — a) 
2 (bod, —b,a,) 2 (bg, — 0, ay) 


if the three points are on the same straight line. 


Ne 1 ay by 1 é 
4 (by a2 — 6, 4;) Cy by 1 
4 Ay(bg— },) 0, b(dy— A) Dg Ay — 0, Ay 


Now add the third column of this determinant multiplied by 
-— a, to the first column; also add the third column multiplied 
by — 0, to the second column. Then 


a 1 0 0 | 
4 (byd_ — 0, a;) Cy — Cy by — b, 1 
1,0;(,— 2) 0, (0, — 02) gg — BO, 


which is obviously zero. Hence F, D, £ are on the same right 
line. 


Itt. To obtain the equation of a circle passing through three 
given points. 
The general equation of the circle is 
(a +y°) +2ax +2by +c=0. 
If (2, 41), (ey Yo), (Xs, Ys) are the given points, 
(a + 9°) + 2aa,+2 by, +c=0, 
(29? + Yo”) + 2 day +2 by, +c=0, 
(5? + ys”) + 2 au, + 2 by, +c= 0. 


APPLICATIONS AND SPECIAL FORMS. SEUE 


These four equations are simultaneous for the parameters 
a, 6, c; hence, by 79, 


ota Qe Qy 1 
m+ yy 2%, 2y, 1 
Wy + Ye 2%, By, 1 (C) 
Ws + Ys) 2%, 2y, 1 


which is the equation sought. 

That equation C is the required equation of the circle deter- 
mined by (2%, Y;), (®25 Yo), (#3; Ys), is Obvious from the form of 
the first member. The determinant when expanded obviously 
gives a function of the second degree, and having the charac- 
teristics which distinguish the equation of the circle. Moreover, 
this equation is satisfied for ~=2,, y=y,, since in that case 
the determinant vanishes. The same is true if r=x, y= yo, 
ee 3, = Yo 


IV. To find the relation connecting the mutual distances of 
four points on the circle. 

We must have, if the points are (4%, 4%), (2, Yo), (3, Ys), 
(2, Ys), a determinant equation just like the last one above, 
except that the first row of the determinant will have the sub- 
scripts 1, the second row the subscripts 2, and so on, the last 
row having the subscripts 4. 

Accordingly, multiplying together 


ae 2 
x MY U+y|> 


ety —2% —2y% 1 
Lh Yo We + Yo" 
oh 
1 


1 
ty + Ye — 2%, —2y, 1 
wy + ys —2% —2y, 1 
Pepe 2% 4.72% 1 


2 2 
Uz, Ys Us + Ys 
: 2 2 
te Ya Re A Uy 


which are two different forms of the first member of equation 
(C) above, we obtain the required relation 


| 0 (12)?. (18)? (14)?;=0, 
(12)? 0 (28)? (24)? 
(18)? (23)? 0 (84)? 
(14)? (24)? (34)? OO 


102 THEORY OF DETERMINANTS. 


in which 

(12)? = (@ — %)? + (%1—Ye)*, (18)? = (@ — %)? + (41 — Ys)’, 
and, in general, (7k)? is the square of the distance between 
the ith and kth points. 


Expanding this determinant by 63, III., and adding and 
subtracting 4 (12)? (13)? (24)? (84)?, we obtain 


[ (12)? (34)? + (18)? (24)? — (14)? (23) 7]? 
— 4 (12)? (18)? (24)? (34)? =0. 
$[ (12) (84) — (18) (24) — (14) (28) ] 
[ (12) (84) —(18) (24) + (14) (28) ]} 
x $[ (12) (84) + 3) (24) — (14) (23)] 
[ (12) (84) + (18) (24) + (14) (23) ]}=0, 
or (12) (34) + (13) (24) + (14) (23) = 0, 


which expresses the condition sought in its simplest form. 


Whence 


V. To find the condition that two given straight lines in space 
may intersect. 
(a) Let C—-a Yy—PB.2—y¥ 
iy ER cr 


; (1) 


OF a is a—YV (2) 


be the equations of the lines. If these lines intersect, the 
plane petqytread 
may be passed through them, and we must have for the first line 


pa i, eee (3) 
py + qh +74 =0 5’ (4) 


and for the second line 


py + 9Ai+ry, =d ; 2 (5) 
Paz + ghy + re, = 0 ” a 


APPLICATIONS AND SPECIAL FORMS. 103 
From (3) and (5) 


P(a—4)+9q(B-fi)+r(y—n)=9 (7) 
(4), (6), (7) being simultaneous, the required condition is 


Cy dj Cy 
Cy Dy Co 
Be a pure a 


(6) If the straight lines are given by the equations 


==\(}, 


aMe¢toaytaz=da ; i 
My + boy + C22 = dy : (1) 
ge + bsy + C32 = ds ; 2) 
4% + Dyy + Cz = dy ; ( 


these four equations are simultaneous for the point of inter- 
section (#, y, 2), and the condition of intersection is 


| a 05 C3, | = 0. 


VI. To find the equation of a plane passing through three 
given points (%, Yr, 2), (Loy Yos 22) (Ws) Yao 23)> 
Let the plane be 
aHe+boy+oqz2=d. (1) 
We must have 
Cy Xy + bi + Ce = dy 
My Xa + OY. + Q%=d >. (D) 
Cy Xz + D1 Ys + 2% = Th 


Equations (D) and (1) being simultaneous for the para- 
meters a, 0), ¢, 4, we have for the equation sought 


| a) Baty i a 
1 Hy 4% 1 
Qs) Ug ego a 
U Ys % | 


104 THEORY OF DETERMINANTS. 


VII. An interesting application of determinants is afforded 
by the following problems. 

(a) To extend a recurring series of the rth order without 
knowing the scale of relation. 

As is well known, a series of the form 


Up Uy @ Ug? oes + UU" oes UL, OY ee 


is a recurring series if the relation of any r+1 consecutive co- 
efficients Un, U,_1, *** U,_, can be expressed by a linear equation 
(the scale of relation). Under these conditions the series is 
called a recurring series of the rth order. Every such series is 
accordingly determined when 27 of its consecutive terms are 
known. If all the coefficients, with the exception of the 27th, 
are known, this last is easily found. By the conditions of a 
recurring series 


Up Pyy 1 + Ply_o + Pylyp-g Fo +Dp ate + Plo =O) 
p41 ee Bie 1 a apse ‘oink + PU = 0 


Uy Ror care ihe orgies EPpy + DP-Up1 = 0 | 
Us, b Pillo,_1+ P2tlor_2+ Pylap—g+ 0+ + Pri t PU, =O J (Ff) 


Now, by 79, 


Up — Up-1  Up-g Up-g o9* Uy Up 
Ur41 U, Ur] U,_9 2 Sie Us Uy 
Unio Uysy Up JA eee 
Udr-1 Uor-9 Uor-g Uop-4 °** Up Up_y 
Uo» Uo, 1 Uo,—9 Uy 3 oe Up4y Uy 


whence w, is found by expanding the determinant and solving 
the equation. 


To find w,4; we have only to increase each subscript by unity. 
Applying the above process to extend the series 


L+a+5e2+ 1308 4., 


APPLICATIONS AND SPECIAL FORMS. 105 
we find 
i aap 8-5 i u, 18 5 
18 5 1/=0; |m 18 5)/=0; |uy wu 138)/=0; 
U 138 5 ie Mois Ue: ee 


whence uw,= 41, vu; = 121, uz=365. The series is accordingly . 


1+e+5a° +1308 + 41 at4+ 1210+ 365 0%+.... 


(6) To find the generating function for any given recurring 
series. 

Since a recurring series is always the quotient of two integral 
functions, of which the divisor is of a degree higher by 1 than 
the dividend, we may find the required generating function by 
indeterminate coefficients, as follows: 

_ Assume the given series 
Ap +A, e+ aoe? +--+» +a,_v" 1 
1+ pe pow? + +++ + pa" 


(after both terms of the fraction have been divided by the first 
term of the denominator). 

From the first r of equations (/’) of the preceding example 
we can determine the constants ,, p....p,- We may therefore 
find the scale of relation. We have from equations (/’), after 
obvious interchanges of columns, 


Unt Uje+ ue? +--- +a" = (T) 


| — Up, Uy 28? Up-g Uy-1 

a Unt Us — Uy,-_} U, 
— Unig Ug ses Up,  Upyy 
| — Ugr_1 Ur Ugr—3 Usr—2 

PD = 

Uo Uy © U,_2 U,—] 

Aiea Bho t,t, 
Ug Ug Ms Up Upt1 
Up_y Uy *** Uny—g Uar—2 


Having determined the constants p,, p....p,, we need only 
elear equation (7) of fractions; and then, equating the co- 


106 THEORY OF DETERMINANTS. 


efficients of like powers of x, obtain the usual linear equations 
from which do, @, d,...d,_, are found. 

For an example, let us find the generating function of the 
series we extended in the last example. 


Put 
; 2 3 i oe Oo, 
l+e+5e+1380?+ 41at+ = Tp etn ne (T;) 
Here | 


Uy = if Oty aa Ye =a) at 


5+ p, +po= 9 


Ce Tes = 23 — —3. 
Fe he okene “ 


Substituting in the second member of (77), clearing of fractions, 
and finding the values of a) and a,, we find 


1—2 
AG = 1—2e—322 
- 85. The coefficients of the quotient Q of two polynomials 
P, and P,, and the coefficients of the remainder 7, can always 
be expressed as determinants in terms of the coefficients of 
P, and P;. 
The method employed in the following example is applicable 
in general. | 
Let Py = ae + a, v* + dy 0? + Agu? + a4” + Os 3 
Py = byw + b, 27+ boa +b; 
Q =H +2 +93 
R=nv+re +7. 


Now P.Q+ R= P,; 
hence Gg 060s 
th = 11% + OH, 
Oba = boQy + 01% + 00 Gos 


(p) | ds = bg dy + bq) + 0140 + 105 


= 6391 + b29o +115 
as bs Qo + 1 


APPLICATIONS AND SPECIAL FORMS. 107 


From the first three of equations (p) we can find q, G1, G2, and 
then taking the first three with each of the others in succession, 
we obtain 7, 7), 7. For example, 


by d= bo 0 Qo} 5 oer Do 0 0 Cg |e 


Dy bo Ay bd; Do 0 Ay 
by dy dy by 0, dy Me 


Let the student find the remaining coefficients. 


86. The coefficients of any equation can be expressed in 
terms of the roots as the quotient of two determinants, as fol- 
lows. The method employed is applicable in general. By 
reference to examples 6 and 7, page 37, it is readily seen that if 


Sf (@) = & — a0" + A,% — ds = (% —a) (w— B) (w—y), 


we have 


1 1 1 1/=—(8—y)(y—¢) (a—8) (e—2) (eB) (e—y)- 

St ae 

ed a? B? y 

a a® B° y 
Expanding the first member, 

MG se |. oe yl leas y| 
a B? y az Bs y B y a B y 
a? 3 y a? 3 7 a? B y a2 3? y? | 

=) a i | 
a By |(«#—a,x + doe — as). 
az B? y 


From this identity the required expressions in determinant form 
are at once obtained by equating the coefficients of like powers 
of &. 


87. With the aid of determinants we readily find the sum of 
the like powers of the roots of any equation, as follows : 


108 THEORY OF DETERMINANTS. 


Let 51, So 83+. 8, Genote as usual the sum of the first, sec- 
ond, ... nth powers of the roots of 


gi + pe" + p.a™—? + eee + Pm1% + Dm = 0. (1) 


Then from the theory of equations we have 


P1 +S; =0 | 

2 Po +718, + 83 = 0 | 

3 Ps + p28 + Pise + S83 = 0 L 
(n—1) py_it Pn—2S1 + Pn—s82+ Pn—s83 +** Spa == 0 | 


NP» + Pn2151t Pa 282+ Pn—383+ sy + PiS,pits,= 0 J (S) 
From equations (S) we obtain at once 


S,= (—1)" Pi 1 O «+» O O}. 
2 Po Pi 1 
3 Ps Po Pr 0 0 


(n—1)Pn-1 Pu-2 Pn-3 + Ps 1 
Pn Pa-i. Paae 2°* Pa ee 


If in (1) the coefficient of x” had been j», we should, of course, 
~) instead 


Po 
of (—1)", and py ingtead of 1, for each element of the minor 
diagonal of the determinant. If n=3, and n=4, the above 


formula gives 


have to write in the value of s, just obtained, ( 


S=—| p, 1. 01, and $=) %p)9 eee 
2p, p, 1 2 D2 i ae 
SPs Po Dy 3P3 Po pi 1 
respectively. 4p, Ps Po Pr 


88. Equations (S) can also be employed to give the value 
of the coefficients in terms of s,, s,, 83... 8,, by solving these 
equations for the coefficients. We find 


APPLICATIONS AND SPECIAL FORMS. 109 


meas 1 Oa 
n! | 8 S} 2 en 
83 82 8] eee, 


Pr 


= 
on) 
=) 


Sy-1 Sn—2 Sn—3 Sn4 °°° oi — 1 
i Sn Sn—1 Sn—2 Sug 32? So 8] 
If, as before, the coeflicient of a in equation (1) had been jp, 
° ° ° er n aed 
we would write in this value of p,, ( :) instead of Ga 
nN: n! 
itetjeso.. and n= 4, 


meee sy: 1h 0 Seige 60-0 
MR cs eT o4le 3) 2° 0 
S83 8, 8 Ser 85? 8... 3 


89. Any differential equation of the form 
Ys + X1Yo + -Xoy, + X3y = 0, (1) 


in which y, ¥;, Y, y3 denote a function of x and its successive 
derivatives respectively, and X,, X,, X; are also functions of x, 
can be reduced to an equation of the next lower order, provided 
a particular solution of (1) is known. 

Let y=z satisfy equation (1). Then 


Za X12. + X_2 + X,2=0. (2) 


cal 
Put U="—-Zy, VSwU. 
Then, as above, denoting derivatives by subscripts, we have 


—v +2zy,—-“y=0. 
— Vy + 2Y_— My = 0. 
“ — Uy + 2Y3 + AY — ZI — ey = O. 


These three equations and (1) are simultaneous; hence 


110 THEORY OF DETERMINANTS. 


XxX Xe X34 =0. 
@ 
0 


Now multiply the fourth column of A by *, then add to the 


fourth column the first multiplied by zs, the second multiplied 
by %, and the third multiplied by ~, and we have 


{1 >. Xe 0 = (0% 


or VUo% — V1 (2 — Xy2) + V (Ss + Xoz%) =0, 


which is a differential equation of the second order. 


Resultants, or Eliminants. 


90. If we have given a system of n homogeneous equations 
containing n variables, or, what amounts to the same thing, 
nm non-homogeneous equations containing n—1 variables, it is 
always possible to combine these equations in such a way as to 
eliminate the variables and obtain an equation of relation be- 
tween the coefficients of the form 


ty a (1) 


FR, when expressed in a rational integral form, is called the 
Resultant or Eliminant of the system. In 77 and 79 we 
pointed out the fact that the equation A= 0 must hold be- 
tween the coefficients of a system of equations if they are 
consistent with each other (simultaneous). In the examples 
of 84 we repeatedly found the resultant of given systems of 
equations. Among the most important problems of elimination 
is the following: to jind the resultant of two given equations, 
containing a single variable. 


APPLICATIONS AND SPECIAL FORMS. 


We consider first 

Huler’s Method of Elimination. 
91. I. Given 

J (®) = pow’ + pix +p.=0, 


and P(&) = Gx + e+ qe. 
If these equations have a common root, we must have 
S(#) $() 
I 2 (ae+a), SP=@,2+0,), 


A ll 


(1) 
(2) 


in which a, d., 0, b, are undetermined, since r is unknown. 


Then 


(b,x + dy) (py @?+ pH + po) = (a4 + Ag) (Gow? + Ge + Ge). 


Whence the equations 


bi Po + 0 —HG+ 0 = (0. 
611 + b2P) — 191 — 2G = 9. 
Dy Po + by py — Go — 2G = 0. 
0 +bopot 0 —A2q,=0. 
Hence, by 77, the resultant is 
R —— Po 0 Of 0 = 0. 


Pi Po NH 
Po Pi We UN 
OFF ps 0" gs 


II. In general, let 


S (2) = pro X™ + pO" + pet os + Dn 1X + Pm = 9. 
p(x) oa Gox™ ae Cites oh Qou"? + ad aE Gn-1% 5 Dn == 0. 


Let + be a common root of (1) and (2), and put 


AP) Oy 2) gD? vee Og 1B A Om = fi) » 


daly fh 


gO) _ b,2"—* + bye”? + ee +B, 1% + b,, = gi(®)» 


112 THEORY OF DETERMINANTS. 


in which the coefficients @,, Qo, «++ Gn, 0,5 Do, +++ 0, are unde- 


termined. Then 
Ai(@) $(@) = i(@) f(@). (I.) 


From the identity (1.), by the theory of indeterminate co- 
efficients. we must have m +7 homogeneous equations between 
the m+n coefficients ), g+**Gm; 0), bo+++b,. Hence the de- 
terminant of the system of these m+n equations must vanish 
if (1) and (2) have a common root, and the resultant sought is 
accordingly this determinant. 

As an application of Euler’s method, take the following 
example. ‘To find the conditions that must be fulfilled when 


I (2%) = pot? + pix? + prt + pz = 0, (1) 
6(®)=HV+n7+qex+q,=0, (2) 


have two common roots. 
If (1) and (2) have two common roots, two factors of f(@) 
must be the,same as two factors of ¢(#). Hence 


(aa+b) (pon? + piv? + pot + Pz) = (cx + d) (Gor? + G2? +9st+ Gs) » 
where a, b, c, d are indeterminate coefticients. Whence 


apy + 9 —cey+ 0 =0. 
ap, + bp) — eg, — dq = 0. 
Cp, + bp, — cq, —dq, = 0. 
ap; + bp, — eq; — dq. = 0. 

0 +bp,;+ 0 —dq,=0. 


From every four of these five homogeneous equations we obtain 
a determinant of the fourth order whose vanishing expresses 
one of the required conditions. Hence the conditions sought 
are expressed by the matrical equation 


Po. Pi Po Pp, Oe 
O Po Pi Po Ps 
MM % Ge Qs O 
0G) Oy aan 


a Ses I 


fiir t45 


APPLICATIONS AND SPECIAL FORMS. 113 


~ _X Sylvester’s Dialytic Method of Elimination. 
92. I. Given 


Po® + px? + pow + p;=0, (1) 
Got + qx + % = 0. (2) 


Multiply (1) successively by a, 2, and (2) by 2°, #2, « Then 
we have the following system of equations: 


Pot’ + py xt + pya? + pa? — 4B 
Pot + px? + pox? + px = 0. 
Got? + G1 a* + Goa? = 0. 
Got + ya? + ox? == '(). 


Gov? + 12" + gox = 0.- 


We may consider these equations linear and homogeneous with 
respect to 2, 2*, aw’, x, a, considered as separate variables. 
Hence 

ie == 15 Ph Pe, ps, 0 | ==0. 
O Po Pi Po Ds 
Go 1, Ge. PO 
OF Oy: ty Oy. 0 
0 0 &@ Hh 


Il. In general, let 


J (®) = py %™ + px" Aa ias + Dm1® + Pm = 9, (1) 
b(®)=QHer-+quet+e-+q,1%e +g, =0. (2) 


If we multiply (1) successively by x, 2---«”", and (2) succes- 
sively by w, x°---x”, we obtain a system of m+ 7 equations, 
linear and homogeneous, with respect to x, , 2°, ---a"*" con- 
sidered as separate variables. From these equations we elimi- 
nate the variables by 77 and obtain the resultant in the form of 
a determinant of order m-+ 2%. 


i 
1 
{| { 


1 


| 


114 THEORY OF DETERMINANTS. 


R= Po Pi Yass Pa Patt Pare oe 
OT pp Pie" Da. Pa eee 
0 0 Po 28% DPu-2  Pa-1 Pn 


a a 0 QO see 
OF S05 Pear en aerate 0 
0 0 do Jal. Qn-2 Qn-1 Qn 


eee e080 +$j.@e8 


It is evident from the form of R that the coefficients of (1) 
enter R in the degree of (2), and that the coeflicients of (2) 
enter R in the degree of (1). 

Cauchy’s Modification of Bezout’s Method of Elimination. 

93. I. Given 


PoX? +p X + pou + ps = 0, (1) 
and We +gyet+ qu+gq,=0. (2) 


Transposing and dividing (1) by (2), we obtain successively 


Po _ Pr ® + poe + Ps 
Yo Gx? + qoe + ds 
Pot +Pi ae + ps 
He +H Go® + qs 


PoX? + Pie + pr Ps. 
He+nextd ds 


Clearing these equations of fractions, we have 


(Po — WP1)® + (Pods — WP2)% + (Pods — WD) == 
(Po0Jo— YoP2) & + [(Pods—YoPs) + (P1d2— NPs) |e+ (1193 —Gi Ps) = 9, 
(PoV3 — Jo Pz) + (2193 — ViPs)& + (Po d3 — Y2Ps) 09 


Eliminating a and w, regarded as distinct variables, from 
these equations by 79, we find 


APPLICATIONS AND SPECIAL FORMS. 115 
K=||po al |Po Ql Po dsl | = 0. 
| Po Gol | Po Qsl +1 pr gal lpr YI 
| Po | lp Yl | De Yl 


The resultant is found by this method in the form of an axisym- 
metric determinant,* whose elements are easily written, as we 
shall show by another example. Let the given equations be 


Pot + px? + pox + pix + py=0, (1) 
and Qo + 2 + qov’+q,7+q,=0. (2) 
We have, as before, 
Po __ Pit? + pk + pat + Dy 
% NX? + Joe + G3 + Gy 
Pot + pi _ P2®? + psu + py 
Wx + Jou” + 3X + Gy (2) 
PoX +1 + Pr _ Pst +s 
Jot? + 9,2 + Go 3% + 
Po® + Pre" + P2X+Pz _ Ds 
OVEnNY+ Gets, Qs 
Clearing equations (£) of fractions, we have 
| Po 112? +] Po Qol@? +] po Gsle +1 po Ws! =0, 


| po Gal@ + [1p Gsl+1 pr Gel Ja? +01 po aul + lpr gs) Ja +1 1 Gl =, 
1% 931@? + [1 Po Gal +1 pi 9s! Je? +01 pi gl +1 po Qs| ]u+| pes ql =0, 
| po Q4l2° +1 pi Q4la? +1 po gla +1 ps qa! =(). 


Hence, as before, the resultant is 
R= |\lpoaul | Po Ql | Po Qs | Po Gal | = O. 
1 Gel |poQsl+lpi Ql lo Qaltlinr gs! lar gl 


109s! = 10 Gal tla gs! lor Galt] pe dsl |p a4 
| Po Ys] | Pi Ql | po al [Ps Gal | 


* For symmetrical determinants, see 107. 


116 


THEORY OF DETERMINANTS. 


To form this resultant directly from the equations, write the 
two symmetrical determinants 


| Po G1 
| Po Yel 
| Po Isl 
| Po Ul 


lPo G2! |Po dsl 1Po Ql], and || pi Ql 
lo dsl po dal pra Gl lpi Qs 
lPo Ql lpi dl Le al 
lpi dal Le Gal Ls Ql 


| Pi qs 5 


| Po Ysi 


formed from the coefficients of (1) and (2) in an obvious and 


easy way. 


It is then evident that R is formed from these two 


determinants by adding the elements of the second to the four 


inner elements of the first. 


If the equations are of the fifth 


degree, the student will form the resultant in the same way 
from the three determinants 


Po Ql |Podel 109s! 190 G4! 109511, 


| Po Jz! | po Gs! 
| Po Is! | Po al 
1 Po Gal | Po0 sl 
| Po sl | pr Qs! 


| pods! 109s! |p: 9s! 
| pods! |pr9sl | pe qs! 
1719s! |p29sl | ps3 9s! 
|p2qs| |p35l | Qs 


lpi 92! |p. 4s! 1a. Gal |, 
| 93! lpr del |p Gal | 
11 Gal 12 Gal Ips Qal 


| D2 Ysl 5 


by adding the third to the middle element of the second, and 
then adding the elements of the second to the nine inner ele- 


ments of the first. 


This process is, of course, general. 


From the preceding examples we see that by Bezout’s method, 
the resultant of two equations, each of the nth degree, is a sym- 
metrical determinant of the same degree whose elements are either 
determinants of the second order or the sum of such determinants. 


II. If the two equations are not of the same degree, suppose 


we have given 


Pot + pix? + py x’ + pz% + py = 0, 


Got +9, % + 


= (0, 


Multiply (2) by 2°; the equations are then 


Pot +p + por + p,% + p,=0, 


Jot? + G2? + Gox? 


=a. Uy 


(1) 
(2) 


(11) 
(21) 


APPLICATIONS AND SPECIAL FORMS. . 117 


From (1,) and (2), 


Po __ Dik + Po® + prx + py 
Yo xe + qu : 
Po®TPr _ Po" + pst +p 
MET au” 


Clearing these equations of fractions, we have 
I Po 11a" + | Py Ga!” — Gops% — GPs = 0, 
| Po Gol a + $1 pi Qe! — qo Ps} a — (GoPs + U1 Ps) *® — Py = 0. 
With these equations consider (2) multiplied by a, and (2), 
HV +Hne + qx=0, 
He+ynr+gq =0. 


From these four equations eliminate 2°, 2, 2, and we have 


R=} lou! | Po Qo Yo Ps Jo Ps| = 9. 
lpoGel | Pid2l—G Ps WPtnHDs NPs 
Yo On —Q 0 
0 % —N —¢ 


IIf. In general, let 
S(&) = Pot” + pw + pow”? +e + Pp e+ Pn =O, (1) 
(2) = Jor” fe g 2 +f qx" a et =a Qn-1 © i i 0, (2) 


in which m is greater than n. Multiply (2) by a”; then 
(2) becomes 


Qo” + Gu" 1+ gum? + AP ue gga, C25 
From (1) and (2;), 


Po Py 8") + poe? + S20 = Dg OE Dry 
J ae Ce dee af Oe ashe au re + ote gym -n+1 of Ree 


PoX + Pi pot”? + pau > + <0 D1 2 A Din 
DEQ qu"? 4+- qa" F 4 --+9, 1:0" ™14+¢q,a0"" 


PoP AEDT PH + FD p- 2 ADna DnB +P np Et FP, 
Qn" * ++ qua”? + soe +O, 20 + On-1 Qu” 


ee i 
, 


118 THEORY OF DETERMINANTS. 


Clear these equations of fractions, and consider with them 
the following m—n equations obtained from (2) by multiply- 
ing it in order by 1, a, a, ---¢™-""}, 

Qoe™ > + QU"? + Goa F4 o> + Gy, 10" + Gem OF =0, 
igo 4. gu" 3 4 oo Dn—9 0m" Ta-10" "are 


Qox” a qa coe + Qn—-1& a Tn =(). 


From these m equations the resultant is obtained by elimina- 
ting the m—1 successive powers of w regarded as separate 
variables. 


The Resultant in Terms of the Roots. 


94. Given 
SS Dy U™ + PU" Hoe Dy 1 + Dm = 0, (a) 
eo) a Go” + Gye + oe = Qn-1% Oy = 0. (b) 


If aj, a, ...a, are the roots of (a), and f), Bo, ... 8, are the 
roots of (6), we have, of course, 


J = Po(% — ay) (% — ag) +++ (% — ay), (c)) 

b = Go(% — Bi) (% — fe) +++ (@— Br). (01) 

Now, if in qa"+q,a" 1+-+.-+q, ,.@+49, we substitute 

successively aj, ay, ... Gm, ~ takes the m corresponding values, 

(a1), P(ag)...d(a,). With these m values as roots we can 

form an equation of the mth degree in ¢. This equation may 
be found as follows. Forming the resultant of 

Dot” py" +++ + Dy 1 ED eee (1) 

Got" + Qa? + +, 1% +4,—-6=0, (2) 


by 92, we have 


APPLICATIONS AND SPECIAL FORMS. 119 


By=| Po Pi Poe Pn Puit Puig ° “se peserUe 
O Po Pree? Dna Pn Puvi 
RANE UN ens na ey pee 
Oh Gs G—d OD 0 
0 % Ns nr m—b 0 
0 0 Gr! Qn-2 Unt In— 


eee e886 


This is obviously an equation of the mth degree in ¢, whose 
roots are }(a,), d(a2), P(as), -*+ P(a,). The absolute term T 
of this equation is the product of its m roots multiplied by a 
factor. 

But from the determinant RF, 


T = (—1)"po" $(a1) $ (a2) +++ b(an)- 


Again, since f, becomes identical with (—1)"R of 92, II., 
when we have made ¢ vanish, we see that 


Fi = po" h(a) b(a2) +++ Pam)» 
In just the same way we can show that 
TP! = (—1)"qo" F(R) f(B2) +++ (Bn) 
and hence, after suitable interchanges of lines, 
a R= (—1)™ qo" f(Bi) f(B2) ++ (Bn) - 


95. These forms of the resultant R may be obtained by 
symmetric functions, as follows: 
ST (x) = Pou” + px” + p,u" + as + Pm—1 8 were = OF (a) 
p(t) = G2” + He? + GO? +e + nit +g, =0. (0) ° 


Then aj, a, -+-a, being the roots of (a), and fj, By, --B, 
the roots of (0b), 


120 THEORY OF DETERMINANTS. 


J (&) = Po (@ — a) (a — ay) ++ (&% — Gm) 5 
(a) = q(@ — Bi) (@— Be) ++ (®— Bn): 


Now, if (a) and (b) have a common root, the product 


F (Bi) F (Bz) +++ F(Bn) = P 


must vanish, since in that case some one of the factors vanishes. 
The same statement applies to the product 


(a1) $(a2) ++: $(an) = P. 


But f (Bx) = Po (Bi — 21) (Bi — 22) +++ (Bi — Om) 5 
se) — ae a1) (Bs — my +++ (Bz — Om) 5 


1B) = ro(Ba oar ay) (B,- ae 3 ta ire Om) } 


also (01) = J (a, — Bi) (01 — Bo) +++ (a — Bn)» 
Hay ae Slee ey Ps) sae (a2—f,), 


Noe = Go (4m— Bi) Coen _g) pie (c= Bn) 


P is accordingly made up of mn factors of the form f, —a,. 
We may therefore write 


P= p," (6, day as) ; 


where 7 has all integral values from 1 to n, and s has all 
integral values from 1 tom. /P is moreover a symmetric func- 
tion of the roots of ¢(«) =0, and can therefore always be 
expressed as a rational integral function of the coefficients ; 
and since it vanishes when f(x)=0 and ¢(#)=0 have a 
common root, and not otherwise, when P is expressed in terms 
of the coefficients, P is the resultant of (a) and (6). In the 
same way 


P, = @" IW (a, — B,) =(—1)”"" qo" (8B, —a,), 


where s and 7 have the same values as before. Hence we may 
write the resultant 


R= (—1)™qu"f(Br) f(Bs)-+- f(Bn) = Ps" (a1) $(a3) ++ 6(am)s (A) 


APPLICATIONS AND SPECIAL FORMS. 121 


for both these expressions are rational integral functions of the 
coefficients of f(#) and d(x), which vanish when S(%) =0 
and $(x)=0 have a common root, and not otherwise, and 
which become identical when expressed in terms of the co- 
efficients. The value of R can accordingly be written 


> a = Po" Go" H(B,—a,). (B) 


Properties of the Resultant. 


96. I. By reference to the forms (A), we observe that the 
coefficients 9, ~i:++P, Of equation (a) enter the resultant in 
the nth degree, and the coefficients q, q,-+-¢, of (6) enter the 
resultant in the mth degree; moreover, we readily see that 
(—1)”"" Qo" Dm” is a term from the first form of the resultant, and 
Po" Yn” is a term from the second form; hence, given two equa- 
tions of degree m and n respectively, the order of the resultant R 
in the coefficients is m+n; the coefficients of the first equation 
are found in R in the degree of the second, and the coefficients of 
the second equation enter R in the degree of the first. 

I. If the roots of (a) and (b) are multiplied by k, R is 
multiplied by k™. Since each of the mn binomial factors of 


+ = po" do” U(B, re a) 


is in this case multiplied by &, the truth of the statement is 
obvious. This result is frequently expressed by saying the 
weight of the resultant is mn.* 


III. If the roots of (a) and (b) are increased by h, the resul- 
tant of the transformed equations is the same as the resultant of 
the original equations. ‘This, too, is obvious, for none of the 
factors of & is changed when both roots are increased or 
diminished by the same number. 


* By the weight of any term is meant the degree in all the quantities 
that enter it. The weight of abc? is 6. 


122 THEORY OF DETERMINANTS. 


IV. If the roots of (a) and (0) are changed into their recip- 
rocals, the resultant R, of the transformed equation is (—1)™"R. 


Putting j= (a) and (b) become respectively 


SF (Y) = PY" + Pm ay 1+ Dn oY” +o HPY+P=9, (ci) 
b(Y)=GnY” KIna yr TH In Yr FY tH=9- (Hr) 


Whence 
eee “n(] - =) = ee Re 
ie Oe B, Os (ay aq *** Om)” (By Be *** Bn)™ 


But 


oa )= Root es re aie hes (= 1)" Qn 
(a) ag n) = Do 5 (Bi Beo+* Bn) = ie 


“ey rie = po" Qo" (—1)"™" (6, LS as) = (—1)"Rh : 


hence the resultant of the transformed equations is identical with 
the resultant of the original equations, or differs from it only in 
sign, according as mn is even or odd. 


97. Of all the methods of elimination given, the dialytic 
method is the most direct. Another advantage of this method 
is that it may obviously be employed to eliminate one of two 
unknowns from a pair of equations, as in the following example. 

Given 

Pot + pix’y + prey’ + pry? = 0, 
He+ney +qy +a, =0. 


To eliminate « we form the following equations: 


Pot + pie y + pox’ y’ + ps xy? = 0, 
Pv +pwry +p 2’ +py =0, 
H+ ney+ (gy +43) =0, 
He + Hey +(qoy? + q3)u = 0, 


Her +ntyt+ wy’ += 90. 


APPLICATIONS AND SPECIAL FORMS. ws 


Whence , po Pry Poy Psy? 0 = 0, 
au py poy? psy? 
Gd HY y+; 0 0 
09 HY y+ Qs 0 
0 0 do ny ay +4 


an equation containing only y. 


98. The same method is also frequently applicable to the 
elimination of » —1 unknowns from a set of n equations, so as 
to obtain a final equation with but one unknown. It will afford 
the student a good exercise to find from the three equations 


ey +a,vz+ta, =0, (1) 
Yy2— 4x =0, (2) 
asxy +ae¢ +a, =0, (3) 


a final equation in y, as follows: First, eliminate 2 from (1) and 
(3), and also from (1) and (2), obtaining two new equations: 
in y and z. From these equations eliminate z, and obtain 


2 
yy? + ogy 0 CsCl 
—(AsYAAg) UA, as gy? + (aya? +2a,a;A,)y+tazae 0 
0 — (AsY+Ag) Uo, As’ Agif? + (A177 + 2Agdslg) Y¥ + Ashe 


ae — A, 
an equation in y of the seventh degree. ? 


99. A further interesting application is found in the follow- 
ing examples, in which three variables are eliminated from as 
many equations. Given 


m+e+-e=0, w=a, #7=b, w'=C. 
Multiplying the first equation successively by 
yy Uo, Vey XUy Ws, 


and substituting from the last three, we get 


A+, X,+ 2s = 0, 
b + 2%, + %,%,=0, 
Cc + 2%; + %%,=0, 


CX, Xo + ba, Xs + AX, HX, = 0. 


124 THEORY OF DETERMINANTS. _ 
Eliminating 2%, %%3, %2%g, 
all O0O/=0. 
igen Raab § oe | 
CoP sa eee 
0 Cy Outee 


Had we multiplied the first equation successively by 


1, %_Xg, Hy gy, Noy 


we should find by eliminating 2%2%3, %, Wy gy 


Batis Ree Wiig 4 4 YP 
1 Qe 
Pe 20 Pa 
Lei a0 
If the original equations are 
Mt+wm+4,=0, xwima, w=d, o,°=6, 


one form of the resultant is obtained by multiplying the frst 
equation successively by ee 


Py Wn 23", 


Then by eliminating 


re: 2 2 2 an 2 2 
Yyy Loy Xgy Uy Mey My My, Wy Nyy = Wy XQ Xp, Ny Ue" Ve, 


and substituting from the last three. 


a, ve, Be V_X3, Xs, 1%, 2" i! a2 Vp ae, EE 
we find 
1 0 0 6° 12120 tea 
0:1 0. ki - Dea 
0 0 1 LS1L -06Gaee 
0: c 0-00. 0a 
c’'@ a © 0 0: Ome 
rb a +0 0 0 O- Grae 
pO: OD a 0-0 Qos 
Oo 0 0 0 2 0 120 eae 
0O 0 2 0 0 ¢ sha 


APPLICATIONS AND SPECIAL FORMS. 125 


100. For a final application of the dialytic method we select 
the following. 


Given Vaot+a, + 1 Cine a Miao 0, 
to free the equation from radicals, we may proceed as follows. 
Put Vay + a, =, Vbyt + by = yp. 


Then we get at once 


wy +Y2 +eo=0, (1) 
Yy — Aye — a, =0, (2) 
Yo — bot — b, = 0. (3) 
From (1) and (3), 
1 0 —hx—b, = 0. 
1 y%+% 0 (4) 
0 1 Y1 + C% 


Eliminating y, from (2) and (4), we have 


1 2g ofF—«x—b, 0 = 0, 
i 2, Co — bv — d, 

a & — Aex— a, 0 

en 21 0 — AX — Ay 


which is the equation sought. 
In general, given 


LCV arava Lah pa. a ges n/ 
DV fil@) + po W/ fa(x) + ps V f3(@) mie +p, In(@) = BR, 
in which 7, 7)---7, are integers, and f\(x), fo(@)-+-f,(@) are 
rational integral functions of 2, we may rationalize the expres- 


sion as follows. Put 


Ai(®) =n"; Jo(X) = Yay %, 22 Sule) = Ya 


126 THEORY OF DETERMINANTS. 


Then we have a system of n equations, from which, together 


with 
PrY~ri t+ PoYo to $DPrijn = KR, 


we eliminate the m variables y,, Y, ..-Y,, and obtain a resulting 
equation in x without radicals. 


Discriminant of an Equation. 
LOL. I. Given 
S (®) = poe” +p 2") + pou”? + ++ +p, 1X +p, = 0, (1) 
_and the first derivatives of f(), or 
Sf! (a) = np" + (n—1) pia"? + (n—2) pat” P+ +++ Ppa (2) 


Then the resultant R of f(w)=0 and f'(x)=0 is called the 
discriminant of f(w)=0, since, if A vanishes, f(#)=0 and 
J'(«)=0 have a common root, and hence f(#)=0 has equal 
roots. 

Forming the resultant of (1) and (2) by 92, we have 


R= 


Po Pi Pe lesa ES Pn-1 Pn 0 0 0 been) 
0 Po Pi ts {Daca Deon Da 
0 0 Po AS ee eee Pn—s Pn-2 Pn-1. Pn Qs 
Mpy (N—1)p, (N—2) Pores 2Dn-2 Pn 0 0: One 
0 pp  (n—1)p;+-- 3p, 3 2), Deo 

0 O 


0 0 NPo ae 4Dn4 3 Dns 2 Dn—2 Pn-1 


in which the first (n —1) rows are formed from the coefficients 
of (1), and the last n rows from the coefficients of (2). 

Now multiply the first row of R by n, and subtract it from 
the nth row; the nth row becomes 


0 —p. —2py+++ —(n—2)p,-2 —(n—-1)p,1 —mp, 0+ 0. 


a 


APPLICATIONS AND SPECIAL FORMS. 127 


Hence R is at once reducible to a determinant of order 2n—2 
multiplied by po; calling this determinant A, we have 


Wiad yay 
Now = B= py" (a1) f"(aa) f"(a) + f"(a,) 3 (94 or 95, A) 
fe) fF) FO), FO). 


and since [Ng fae age ears x—a. 


n 


J' (a1) = Po (a, — ag) (a, —=itte } feck: (a, ete) (a, — a, | 
J" (a2) = Po(a2 — a1) (a2 — ag) v2 (Gg — Gn_i) (a2 — ay 
f(y) = Po(4n—1— 41) (yaa) #** (041 Oy -2) (yay) | 

J" (an) a Po(Gn— 21) (4, —az) xe (a,,—an_2) (a,, 2,1) } 


If we multiply equations (#) together, we see that the second 
member of the result will contain the product of the squares of 
the differences of the roots a, a,,...a, of (1). Employing the 
usual notation for this product, viz., €(a,, a2, a3,+++a,), we have 


SF! (a1) f"(a2) vos f' (aq) = aa py oo; Gg) gy *** Gn) 5 
nA (—1)20- p22 C(ay, Dg ree ae 


Il. The discriminant of an equation can also be obtained as 
follows : 


f(#)=9, C1) and f'(#) = 9; (2) 


being simultaneous equations when /(x)=0 has equal roots, 
the equation 


nf(x) — af'(a) =0 (3) 


is also consistent with (1) and (2). Now (3) is an equation 
of the (n—1)th degree; and finding the resultant of (3) and 
f'(«)=0, which is also of the (n —1)th degree, we obtain the 
discriminant A as a determinant of order 2n—2. For an 
example, we shall find the discriminant of the cubic 


PoX? + PL? + Pr® + pz = 9. 


128 THEORY OF DETERMINANTS. 


We have to find the resultant A of the equations 


Pe + 2p.%+ 3p; = 0, 
Spot? + 2pyV+ Po = 0. 


A= Py 2 Po 3 Ds 0 = 0. 
0 Pi 2p, BPs 
3 2P1 Po 0 


0 39 2p, Pe 


By the same process we find the discriminant of the biquad- 


ratic P=px't+ 4p, + 6p.2° +4 p,x +p, = 0* 
to be 
A=| Pp) 3), 3p. Pps 0 0 |=9. 

O° py 8p; She> Pp, 0 

0 0 po 8p, BP. Ds 

Pi BP. BPs ps 0 0 

0: py -3ps- 3ps- Ps BV 

0 0 Pi Sp. 8p3 py 


This is accordingly the same as J?— 27J°=0, where 


I= pops — 4p p3 + 3 po’, 
SJ = Po Pops + 2 Pi PoP3 — PoPs — Pr Ps — Pa’ 


102. We may show that J=0 is one of the necessary con- 
ditions when the biquadratic P= 0 of the preceding article has 
three equal roots. Since 


P=px'+4pie+6p.2+4p,0 + p,= 0 (1) 


* In many processes it is found more convenient to write a given func- 
tion in the form of this equation, ‘.e., 


Poe” + mpy wrt = (n—1) pyaeh-2-4 3 (n= 1) (n—2) pyar + 


+ — (n—1) pn—2x? + npn—1X + Pn, 


iS 
2! 
in which each term is multiplied by the corresponding coefficient in the 


expansion of (v+1)". Any given polynomial can, of course, be at once 
reduced to this form, 


4 a 


APPLICATIONS AND SPECIAL FORMS. 129 


has three equal roots, two of these will be roots of 


Pe+3 pe + 3p.% + p;=0, (2) 
and one of them is a root of 
Poe’ + 2p, u + p,= 0. (3) 
From (2) and (3) this root is also found in 
Pe+2p.e+p,=0. (4) 
Multiplying (3) by 2, (4) by 2a, and adding, we obtain 
a? (py 2? + 2p, % + po) + 2x( pe? + 2p.e + p;) = 0. (5) 


Now adding p,2’? + 2p,%-+p, to the first member of (5), we 
have, since P= 0, 


a" ( pox? +2 pe+ po) +2 a( pa? +2 pret ps) + pox? + 2 p+ py= 0. 


Hence, if (1) has three equal roots, 


PY +2pe2+p2=0, | Po Pi pol = 9, 
PX + 2p,% + ps = 0, Pi Pe Ps 
Dot + 2p,% + p,= 0. Pe Ps Ps 
or eet), 
The other condition for three equal roots of (1) is accordingly 
aa 


103. The resultant of a system of n homogeneous equations, 
one of which is of the second degree, and the remaining n —1 
are linear, may be obtained as follows. Given 


PE PV + py + pre + 2Qxy + 2nwz+2Qnyz=0, (1) 

PH=aqe+hy +q2=0, (2) 

Pr= dye + doy +z = 0. (3) 

Differentiating (1) with respect to x, y, z in succession, and 

remembering Euler’s theorem on homogeneous functions, we 

obtain 

P= A P+ HIYAUZE)AY (e+ AY + G2) 

+ 2(H2 + Gay + p22) = 0. (4) 


4 Z 
“a 


130 THEORY OF DETERMINANTS. 


Equations (2) and (3) and (4) are simultaneous homoge- 
neous equations ; hence, by 77, (4) must be expressible linearly 
in terms of (2) and (3), and 


6, P, + 6,P, = 0 (5) 


is an equation identical with (4). Equating the coefficients of 
(4) and (5), we have the following system of equations: 


P+ HY +He — 44) — O20, = 0, 
Go + Pry + G% — 4,0, — 6.6, = 0, (E) 
1% + Qoy + poz —6,¢ am 65 Ce = (Q. 


Now, taking equations (2) and (8) with equations (2), we 


have a system of five homogeneous equations. Eliminating 
X, Y, 2, %, G2, the resultant of (1), (2), (8) is 


H=1D % Hh th |. 
Gd Pr Ge O Ob 


M1 G2 Po GG C% 
d, Dy ic,. 0. 90 


In general, let the system of equations be 


S(@) = peP + pow? + psu? + + + Py Xp? + 2 Gy) %e 

+ 2 Go X,X3 + +: +29 (ny nay a iP 
Pi =X, +6,% +6% ++-+ha, =0 
Ea ee 
gts = Ay-1% tz b,-1%¢ + Cy_1 V3 2 “pes “fe bacy tee a J 


We have, as before, if fy,’ denote the differential coefficient 
of f(x) with respect to 4,, 


Wy fr,' + Xo fxg! + Ws fr! oe 4 tn fx,’ = 2f (2) = 0. (by 


Since (a) and (b) constitute a system of simultaneous 
homogeneous equations, (0) considered linear with respect to 


ae ta 


APPLICATIONS AND SPECIAL FORMS. 131 


the variables, must be expressible linear ly in terms of the 
nm —1 linear equations of (a). Hence (0) is identical with 
0, P. it O2 P. + 63P. gt ~ + O,3 re 1 = 0; (c) 
Equating the coefficients of (b) and (c), we obtain the n 
homogeneous equations 
Dh FM La+ Jog +++ + Yn 1 Vy= AO, + Mob + gbg +++ An Ona) 
NX HF P2Vot Quist +++ + Jon 1@p = 019, +0062 + 0303+ ++» + On 1On—15 
QoX, + emer © HF Ysn—33 = oie. os vent wie es bee Ly 


Gn 1% + don— 1% + U3n— 3X3 +: pat aaa 18, af 1. + ct Fe 7+ bo pak 


These equations, together with the n —1 linear equations of 
(a), form a system of 2n—1 equations between 2, 2, --- x, 
6, 92, ::- 6, ,. Hence the resultant of the given system is 


Pin, Yi Mie a Uned ey. tg se Oy ts 
M1 pe Goes) Gita, 015 02) + On.) 


Gs . Un Det wn Cis Ca et Cel 


eee e880 e828 


Qn—1 Yon-1 Ysn—3 **° Pn l, ls, Ba tna 


Qy b, Cy hand i 0 0 
a2) b, Cg E 0 0 0 
yi Dod Es Peed se lia 0 0 hie 0 


Special Solutions of Simultaneous Quadratics. 


104. By the help of a special expedient we may often solve 
a pair of simultaneous quadratics much more rapidly and ele- 
gantly with determinants than by the ordinary methods. The 
following examples will serve to exemplify the method em- 
ployed, and are, moreover, such forms as occur frequently. 
A. Find x and y in 
MU+tdyy mM 
a0 Oyo. mys (1) 
et par 


tty a Sees tacos 
: : a sk ea 
ee oe 
$32 THEORY OF DETERMINANTS. _ 


Let f be such a factor that 


ae + dy = fm i ‘ 
Ag + doy = fins 


From (2) 
My by a My, 
PUP Re NES Fe __| Ga teal ee 
To, bel AG | dy A ee A 
Substituting in the second equation of (1) 
A 
2D + fDi=arA; .«. f=—— 
= i VP TD: 
._ t= Lt Le . Y= ere ye 
£/D?+D;’ t/ DTD; 
B. Solve the equations 
Oe + by = mM, LY (1am 
yt + boy = My xy a 
Divide these equations member by member; then, as before, 
put a 
ae (2) 
Aye + boy = firy . 
ei nt _ FIG ml, 
a | Ay b, |’ | ay bg | 


From the first equation of (1) 


ie [a | my bo | +21 ay msl] Lay by! 
m, | my be || ay Msg | 


La, dy |. lay b, | 


| m, be | 


4 . 


| ay Mz | 


a4 


A shorter solution is obtained by dividing each equation of — 
(1) by wy, and solving for oS aiid 


APPLICATIONS AND SPECIAL FORMS. 1383 


C. Solve the equations 


ar +bhy =m (1) 
en pean | 


Write these equations 


He +hy =m, 
(2) 
Age 2+ boy + Y= My 
Mm, by a mM 
My boy Ug% Meg hs <b 
iA Gk Eee CE —_ —___—-, A= : hols 
Then 2 i ere y mn ( dye Bey ) 
We have 
vA — Mm, boy — MyDy, 
My Ag% + Ay = Cy Meo 
Ay b,x — a boy — . 
Hence 
A —m™,b, My by = 0. 
My Os A — A, Mo 
Cy Dy —h be A 
From which re te 
A — + Va,? Dy Mo ok b? Ag Ms a My? de Do. 
Again, 
a¢e+ by =m, 
,0,2 —— Ay boy — BS 
m, 0, a Mm, 
eee: PA aba! peace (a= 4 by :) 
iat A, ; A, E Ag Di — by 


D. Solve the equations 


2+ by =m \. (1) 
Aye + boy + CoxY = Mo 


These equations we write 


M2 + by phen : (2) 
(M2 + Coy) @ + boy = Mg 


1384 THEORY OF DETERMINANTS. 


Ay my 
| m, bg | . ia Ag+ CoY Mg nee Ay by ) 
ag A he A | dg ey bz 


As before, 


(A+ My C2)Y¥ — A M_+ MA, = 0, 
— b, Coy + ly dy — a,b,—-A=0. 


Whence 
A+ Mm Co My42—%mM, | = 0, 
— 0, Co by — gb, —A 
a quadratic from which A is found. 
Lang De" | | Ay Mz | 
es y = ——* 
A A+ M,C, 


Example B above can also be solved by the method of this 
example. 


E. Solve the equations 


aw’ + bey + cy? =d ; : (1) 
ex? + fay+ gy =h 


Equations (1) may be written 


(2) - 


by easy reductions. We introduce the factor 2 for convenience 
in calculation. A solution analogous to D could be given. 
Whatever the coefficient of xy, it can, of course, be at once 
reduced to the form 2a,. We write equations (2) 


w+ 2aay+ by? = m 
w+ 2a,xy + boy? = Me } 


a(e+tay)+y(aqxe+ hy) =m 
% (a + day) + Yy (Me + boy) = Me f. (3) 
Then 
Mm aetby LC+HY M 
Me Aye+ doy L+ AY Ms 


A ’ y ie A ’ 


4 _ APPLICATIONS AND SPECIAL FORMS, 


where 
q A= 


L+MY HMethy 
L+UY Agut boy 


_ We have 


[im — m,] @ + []a,m.|—A]y = 0. 


Whence 
a Fee | by Me | WIE 
, My — My la;m.l—A 


Solving this quadratic, 


(oe +-+/| Gy My|7 + | by Mz | (m— Mz) » 


4 | : 
Now >; aca . 
‘a A+ | amg | 


Substitute this value of « in the first of equations (2), and 
we have | 
4 | my be|? ¥? 2a, | my, by! 7? 
- 4 (A+ | a M\ )? A+ |amy,| 


+ hy =m, 


a8 pure quadratic, from which the value of y can be found 
at once. | 


105. To the solutions of the last article we add the follow- 
ing, in which one equation is a quadratic and the other is a 
— eubic. . 
Find the values of x and y in 


w+ ay ty? _ m 
a a? — ay + y" My | (1) 
i ety = at 
. From the first of equations (1) 
a w(@+y)+y-y = dm \. (2) 
7. a(a—y)tyy = Am, 


136 THEORY OF DETERMINANTS. 
| FY ery My, 
Mg Ye am ‘\a—y ma| Mgt a=[?t¥ 9 — 2,7.) 
ae A Coa ieee eae e—y ¥ 
We have 
vA — Xr (m,— My) y = 0 i. (3) 
Aw (Mm, — My) + [A —A(m, + me) ]y = O 


Whence 
A —— r (m — Mz) = 0. 


A(Mm,— M) A—A(Mm + Me) 


From this equation 
aA=3 [my + my: +V10m,m — 3m? — 3m]. 
2 1 2 § Wat hb 1 2 


Now, since A= 27", 


we have to find the value of » in order to complete the 


solution. 
From equations (2), and the second of equations (1), 


et+y= — 
AMg , (4) 


From equations (4), and the first of equations (2), we get 
a’ 
and hence 
y= 5 | (ust ms) V0 ee 


V4 ( 3 My, Me baa Ms?) 


a2 may be found from the second of equations (1), or from 
the first of equations (8). 


APPLICATIONS AND SPECIAL FORMS. 137 Bs 


Solution of the Cubic. = 

j { 106. The general cubic equation 

Do® + PX + por +p; = 0 (1) 
is always reducible to the form 

- a + 1% + qo = 0. (2) 


We are therefore only concerned with the solution of (2). 
The determinant equation 


A=|e% MQ a,|=0 


q dg © 
, Gy aes 

_ is identical with 

‘a a — 8a,a,7 +a, +a2=0. (3) 


We have 

A=|%+a+4 Ay A]; 
C+a +d CU 

a G++ My A, & x 


hence e+a,+a,. is a factor of A. : Ibi 
‘2 ; ~ = + , Be 
Again, let a be one of the imaginary cube roots of unity; ak 


‘then the other is «2. Substitute (1g, da” for a, and ay re- 
p spectively in A, obtaining 


Al=| @ Qa Goo?|=2—8a,a,0°r+a7o0°+a%a=A, 
ly a” Ay a,a ; 
Aya G07 vw 


7 by . a 
since a’=a°'=1. Whence Soe 


)/— (A=l|a+aa+aa? aya aya’; 
hot ( x + dja + A, a” x Cy a 
, | etaatds,c 1 ONE CS 


138 THEORY OF DETERMINANTS. 


and hence A is divisible by «+a,a+d,0°. By substituting 
a,a2 and dga for aq and a, respectively in A, we obtain a 
determinant A’, which is shown equal to A in the same way 
as before. 

Hence a,c? +a,a+2 is also a factor of A. 

Accordingly, 


A = k(a+a,+ dy) (© + dha + aya") (@ + da7 +20), (4) 


where k is a numerical factor. Comparing the term a of A 
with the term 2° in the second member of (4), we see that kK=1. 


+, 8 — 8a, doa a3? + a? = (@ + Gy + Ae) (& + aya + Aza”) 
(@ + da" + da). (5) 


From (5) we have at once 
; 2 2 
x= — Ay — Ags — a — Aga . OL — doa. 
Now applying this result to the solution of (2), we put 


; 3 3. 
i= — 3G, Mg, Yo = Ay + Ap" 5 
whence 


eS _oooo 
=| 2 ae", a — «| 22 qo 

oe \ 2 + mae af 2 i 
Hence, finally, the roots of (2) are 


4 


Rte Us Gest OF \\ qo V2 qe 
ph Ne +97 tN TS Na 

BES AE cee Mena v= ee vo. (# ag eae ig te 
\- 2/8 a - JH 42 Ht Ne N —': ~ 4242 a ve 1 


APPLICATIONS AND SPECIAL FORMS. 139 


Symmetrical Determinants. 


107. When we regard the square of elements that make 
up a determinant, it is natural to inquire what special proper- 
ties, if any, the determinant possesses when we suppose the 
elements not all independent; in other words, what special 
forms arise when we suppose certain relationships to exist 
between the elements, and what are their most important prop- 
erties. Among the special forms very frequently met with, 
especially in Geometry, are the Symmetrical determinants. The 
symmetry here referred to is first, symmetry with respect to 
the diagonals, and second, symmetry with respect to the inter- 
section of the diagonals, t.e., the centre of the square. Two 
elements, so situated that the row and column numbers of the 
one are the column and row numbers of the other, are called 
conjugate elements. Evidently the line joining two conjugate 
elements @,, and a,, is bisected at right angles by the principal 
diagonal. If in a determinant a,,=a,,, then the determinant 
is axisymmetric, or simply symmetrical. The definition of a 
symmetrical determinant is extended so as to mean symmetry 
with respect to the secondary diagonal also, so that a deter- 
minant is symmetrical if for each element there is an equal 
element so situated with respect to its equal that the line 
joining the two is bisected at right angles by one of the diago- 
nals. The following are symmetrical determinants : 


Om OF Gq dhl, My, Uo Ag Ay ty m% Ob G& Al. 


b, by Cc dy Ge Ag As Ang Ay 0p CG 
Sec CG, Os Cia Gi) Gegae es iy acUge 704.) Oy 
d, d, ds dy! yg, eg gy Ny A A, Ag 


108. We have already had a number of problems which 
gave rise to symmetrical determinants. The student may refer 
to the last determinant in example IV., 84, to the first deter- 
minant of 84, VII., to the form of the resultant obtained by 
Bezout’s method of elimination, 93, (I.), and to the value of 


140 THEORY OF DETERMINANTS. 


J, 102, for illustrations of how symmetrical determinants 
occur in practice. Again, we have 


Zieee 
[eee lea” ag! = 


2 2 2 
Ay + Ay + Oh qa + Ay Mog AygMog — Ay gy + AyQAg9 + Ay 3M%33 | 5 
2 2 2 
(bg 14) + AgeQy9 + Ao3Qy3 Coy + Ogg + Aeg C131 + Age Agg + Ang 33 
2 2 2 
Ag) Ayy + Ageia + Ag3A13  Ag1 Aa + AgoAo9 + Ags Mog As) + Ago + Asg 


which is obviously symmetrical. It is easy to show that the 
square of any determinant is a symmetrical determinant. Let 


| Ay, |? = | Oia! 
then we have to show that b,, = 0,,. 


bs = 4,1 As) as Cb,.9 Aso a Ag Asg Rte st = Ayn Aen 
he = 44,1 oe Aso M9 we Ass Uy3 “t bh he =F Ven Urn 5 


whence the proposition. An obvious corollary is that any even 
power of a determinant is a symmetrical determinant. 


109. It is evident that conjugate lines (a row and a column 
having the same number) in a symmetrical determinant are 
composed of the same elements in the same order. Consider 
now two minors M and M&M, of any determinant such that the 
rows and columns erased to obtain M are the columns and 
rows erased to obtain 4. Then 


M= |\Q,:, G, --|, and M, = )a7 a7 ee 
Aggy Con Agi Hs ng ng Sad 
eee eee eee eee iy Cig tee 


Now, if the determinant is symmetrical, so that a,, = @,,, 
we have M= M,, and, in particular, A,,= A,,; or, in a sym- 
metrical determinant, conjugate minors are equal. From this it 
follows at once that the reciprocal determinant is symmetrical. 
Further, it is evident that minors whose diagonal lies in the 
principal diagonal of a symmetrical determinant (coawial minors) 
are themselves symmetrical. 


APPLICATIONS AND SPECIAL FORMS. 141 


110. We may show that the product of a symmetric deter- 


minant by the square of any determinant is a symmetric determi- 
nant, as follows: 


Let | a,,| be a symmetrical determinant, and put 


| ay, x | ay,,| = | Cra t's and | dy, | x lay, 1? = | By, |- 


Then 
by = jy Cy HF ig Cyo + yg Cyg + vee Qin Cyn (1) 


In (1) substitute the values of ¢. Co, ++ Cm, and we have 


bn = (ay Oy $F Ayy Oye + Aygayg ove? Ayn Ain) Oy 
(hay ayy + Ao9 Oy $F Clog Ayg FE +** +E Clon Ayn) O49 
H+ (Anrty + Ansax2 + Angag Foes + Onn Qin) Xins 


= (Gy G41 + Ag aig + Agra + +++ + 4,4 Cin) Oy 
H+ (yy. a5) + Mog yg + Aggy agg + ++ + Ans Cin ) Ong 
ae (Qn Fan 4g + Agndig + +++ + Cnn Qin) Bens 


Since a, = d,;, this sum becomes 
Ox) Ca = On9 C39 = Peak + Ain Cin = Duss 


Whence |0,,| is symmetrical. 
From this and 108 we see that any power of a symmetrical 
determinant is a symmetrical determinant. 


111. Cauchy’s theorem for the expansion of a determinant, 
example III., 63, assumes a somewhat different form when 
the determinant is symmetrical. Thus, instead of 


/ 
A = Ap A — SA Aq, Aus 

we have, when A is symmetrical, 
A a Clog Ai DOs —— 2 Bs ee Pre Pe 


in which, as before, ¢ has all integral values from 1 to n, and 


142 THEORY OF DETERMINANTS. 


fel 


for ik we write the different combinations of the numbers 
1, 2, --- m, taken two at a time. 
For example, 


a h g|=abe—af? —by’ — ch? +2 foh. 

( Seet Lale 

g f ¢ 

0a b cl=@fP?+h)9+ ch? —2abfg —2acfh — 2 begh 
a 0 h g| =(af+bg—ch)*? —4abfg. 

bh 0. ff 

ie eas § a | 


112. Consider the determinant 


A=la, 0; G& Gh &f, 
Oy Dg Bg Uda 65 
GS (Cha tne: Gare. 
d; dz dgv dy @ 
€; € e : 


and suppose that 


a+b, +e, +d, +e; = 0) +b.+C.+dy+ ey = C+ Cg +C3+d3+ es 
= dht+de+dz3+dy+e, = ei: +eotest+ey+e; = 0. 


Then, first, A=0; since, if we add the elements of the 
other rows to the corresponding elements of the first row, the 
elements of this row all vanish; and, secondly, we can show 
that all the first minors of A are equal. 


B=—-|h & dG &|, and C.=—-\|aq, G dG ah 
co ae at ae 61 Ge Ge. ee 
gms ty — 6s d, dy dy & 
Cy Og") €g- 1 Se") CQ, €5 Cy oe, 


The first, third, and fourth columns of B, are identical with 
the second, third, and fourth rows of C,. By hypothesis the 


APPLICATIONS AND SPECIAL FORMS. 143 


~ elements of the first row of OC, are respectively —c,, —¢3, —ds, 
- —€3; whence ~ 
| Cy =| Cy dg és|= B, 
OintGy, 1g ee 
d, ds dy & 


@ 3 & @ 


7 


as was to be shown. 


In general, if in a symmetrical determinant the sum of the 
4 elements in each row is zer o, the determinant vanishes, and all 
_ the first minors are equal. 


Let 
A = | Ay My eg *** Analy With d,, = d,,, 


and Dig + My + Aig + os + Ay, = 0. 
That A vanishes is obvious. Again, 


ag Oy) yp eee ip CN, Uy p.4-1 ie MO ER 
Co) C9 s* Om 1 Cy, Choy 41 “°° Con 
Hen G19 °9* Uap. Ue UW_1n41 °° Gin 
iy Cig per Cheesy Qin Wins ave UC. 


Qisn Asia 29° Apsray-1  AUpeae, AUtiesr *°° Ustin 


On} Ane s8¢ Any—1 ny Ank+1 *2* Ann 


To the ith row of Aj add the remaining rows; the ith row 


becomes , 
— 1 Ag °° —Aop-1 —Ap, —Aoxt1 ***° —An- 


Then to the kth column of Ay) add the remaining columns ; 
the kth column becomes 


—Ayy —Ag *** —A;_190 A —Ait10 *°* —Ano- 


_ Now, making the ith row the first row, and the Ath column 
- the first column, we have ~ 


> ogee 


144 THEORY OF DETERMINANTS. 


(—1)'t*| Go = An ag 
Ayo yy Che 
4-109 Gn A-_w 
Mii9 Ain Ajrre 


cl nO Oni Ong 


which proves the theorem. 


113. If w be subtracted from each element of the principal 
diagonal of a symmetrical determinant, we have a function of 
« which, equated to zero, gives an important equation. The 
roots of this equation are all real, which may be proved as 


follows. We have 


S(®) =| ay — 2& Ayo 
Coy Aga — & 
Cay sg 
An} Ang 
Then 
J(—%) =| ay +2 Ayo 
Cg) gg + & 
Cgy Ago 
Oni Ang 
Multiplying (1) and (2), 
S(#) f(—2) 
= | Pu—2 Piz 
Pa Poe 
Ps Ps2 
Pri Pre 


Dsg— 2? +++ Dan 


= 0. 

(1) 
Uys = Wey 

(2) 
Us = Dey 
= 0, 

(3) 


Prs = Per 


APPLICATIONS AND SPECIAL FORMS. 145 


where Pix = Uy Ua 1 Ug yg + +++ + Cin Aan 
Expanding the determinant of (3) by 63, L., 
| Pin ie =D, ar v3 Dn -2— 0° D,,_¢ at la? Ae (—a?)” = 0. (4) 


Now D,1, D,-2, D,-3, +++; being coaxial minors of | DPin|s 
are all sums of squares of minors of |«a,,|; for consider one 
of these minors 


D2 = Py Pry °** Pr 
Pot Pog °°* Por 


Pry Prg 899 Dry Pa = Peg: 
D,_». may be obtained by squaring the array 


i ' Op 7) Chr oes Cin 
; Way ga lh gg 4" 6) Cons 


Gry Arg yg 29° Anny 


a 


in which there are n columns and n—2 rows. By 58, Ist, 
D,,-. raust be the sum of products of pairs of determinants 
which in this case are equal; hence D,,, is the sum of squares 
me eminors Of |a,,,| of order n—2. Hence SD,_,, SD,-», 
SD,,-35 +++, are all positive. The signs of the terms of (4) are 
therefore alternately positive and negative, and, by Descartes’ 
Rule of Signs (4), can have no negative roots. Accordingly, 
f(x) =0, or (1), cannot have a root of the form aV —1, for 
then x would be negative, which we have shown is impossible. 
. Nor can (4) have a root of the form B+aV—1; for if we 
write @,—B=a'y, Gye—B=a'y, etc., the proof just given 
is applicable. 

The student will find it interesting to apply the preceding 
proof to the particular case where f(x) is of the third degree, 


U.Eny J (%) =/a)—2 Cys Cg = 0, 


Cg Cyn -F Cog 
Cy Cg3 (tgg Ar L | Ons = gy 


146 THEORY OF DETERMINANTS. 


actually multiplying f(%) by f(—#), and expanding the result 
to obtain the equation in 2’, whose terms are alternately posi- 
tive and negative. 


114. Symmetrical determinants of the form 


Ay Do Cz dl, és 9 and Cp Ay Ag ou An—-1 : 


to ct by Os TR 


lg Ihe OF So G 
An 1 An Unt1 s* Aono 


= P (Ay My*** Agn_2) 5 


are called orthosymmetric or persymmetric. That is to say, 
when each line perpendicular to either of the diagonals has all 
its elements alike, the determinant is persymmetric. Such a 
determinant can contain at most 2n—1 distinct elements. 
Examples of the occurrence of orthosymmetric determinants 
in practice are found in 84, VII. 


115. The most important property of orthosymmetric deter- 
minants is that the determinant remains unchanged when the 
jirst terms of the successive orders of differences of its 2n—1 
elements are substituted for the elements themselves. Consider 
the following series of numbers, and form the 1st, 2d, 3d, 
-++ (2n —1)th orders of differences by subtracting a,_, from a, 
throughout. Then adopting the usual notation, we have 


A; An Ais Ais Ay ihn 3 A, Qn—-8 
A, An Avo Aos ea A» 2n—4 


at f . i 


APPLICATIONS AND SPECIAL FORMS. 147 


, We now show that at 


= Ao Ay Og As eee An—1 — A A, A, A; eee Seret H 
M% MW Ag Ay *** Ay 4, A, A, Ay «+ A, 
Gp Ug Aq Ag +e? Ans A, As, Ay As + Any 


An—1 An An+1 Anz2 °** Agno ja deere ate Vers Aig *** Doge 


If in A the (n—1)th, (n—2)th, --- column be subtracted 
P from the nth, (n—1)th, (n—2)th, --- column respectively, 
we get 
Skim ts han Ay Ay sercbkgse oot 228) Ty 

M An» Ay Ay ses Ayy-y 

de Ay Ars Ay - Ay 


Gyaa Mer Aig Ayia <7 Arons 
Repeating the operation successively, we obtain 
. 


A=|% A A, As; abe sNeRe) , 
ay, Ayn Ay Ag Sis eae 
A Ay Ag Ago pet CES y 


Un—1 A; n—1 A, n—1 As ed °F A,-1 n—1 
Operating in a similar manner upon the rows, we get 


A=|@ A A, A; tees es ’ a 
it ea" ks A, A, vee AL 
A; As A, A; ts An+1 
An1 4A, Ans An+2 ses Aono 
ay. as was to be shown. 5 
Thus 4 
8. 8 hee 226° f= 3 Ber 
8 15) 2p. 43 5 
2 
2 


15 26 43 68 
26 43 68 103 


ae i i | 
oO }W wb 
(SAG ae FO) 


148 THEORY OF DETERMINANTS. 


for we have 
8:85 15 26: “480 (65a oe 


Sea LL Tee 25 ae 
Dopod 2.6 S05 Te 
De eee 2. rf 
Similarly, 
7 0 —4 —5|=| 7 —7T 3°90 Sas 
0 —-4 —5 —38 —T 382 70aae 
—4 —5 -—3 2 3 0a 
—5 -—3 2 10 0 0 Oa 
The student may show that 
EE Soo 0 SSN) en GS Raed Rcree | 
View LZ Lh ee 1 
AT ae 19 1. O° 2 hee 
lou 10s 230 0 1) —2 ee 
* 
Vee AS Coe PaO. 1 8 27.. 640 =6_ 
Ea Ne yt 8 . 27). b4el tae 
0. LEGe Zo web 27. 64 12052 io 
16 25 386 49 64 125 216 3438 


Besides exhibiting obvious simplifications, these examples 
show that when the elements of a persymmetric determinant 
of the nth degree form an arithmetical progression of order 
m* <n—1, the determinant vanishes ; and if the order of the 
progression is n— 1, the determinant reduces to an nth power. 


* The series of numbers 
1 8 27 °647 125 V2i6 
form an arithmetical progression of the third order, because the terms of 


the third order of differences are alike. 
Thus 
1 8 27 64 125 216 
TOLD. Birt Ol eenoL 
12) Le a 
G76 6. 


_ 


APPLICATIONS AND SPECIAL FORMS. 149 


116. The conditions of the last statement will always be 
fulfilled if a, is a rational integral function of k of the mth 
degree, whose highest term has the coefficient 1. For then, 
according to the well-known theorem, qd, @, a, +++ form an 
arithmetical series of the mth order, of which the mth dif- 
ferences will be m!. If, then, m=n—1, all the elements of 
the secondary diagonal will be (n—1)!, and all the elements 
below it will be zeros. Whence the determinant equals 


(—1)i [@—). 


If m<n—1, the determinant of course vanishes. In either 
case, instead of do, a), M2, +++, we may write 


iy W415 Aro, °° 


If, for example, p is any given number, and 


m m } 
pm pt+tm+i 4 ey 
m ( m m 
pt+tm+ 1 tes ) 
m m 


‘p+ 2m pee2m+1 ie Ce 
( m m m 


= (— 1) z+) a (—1) D3: 


. eee 
—— = ’ 


117. Consider the determinant 


A= |k bee re lee | 
ler kr kre eee de® 


150 THEORY OF DETERMINANTS. 


whose elements are in geometrical progression. That A must 
vanish is obvious at sight; for dividing any column except the 
first by the ratio r, A is seen to contain identical columns. 
Hence if the elements of a persymmetric determinant form a 
geometrical progression, the determinant vanishes. 


118. To the results of the last article we add the following. 
Suppose in 


| An—1 Ay, Un+1 Pty: Con—2 


each element divides every other element whose subscript is 
higher than its own, 7.é., in general, 


dh, = 05610, -+= b,. 


Then 
A= bo by dy by, by bob, bob; 
by}, bs by by by bs by dy by bs bg by by babs bys 


By By Do+++ By Oy dy Ba++, Ody bg*+* On 41 000, bg ++* Ons 
By by bg +++ By _y 
by by ++ Bb, 10, 
tion eee Dee 


bo b, a Don—2 


Now it is obvious that 0) is a factor of the first row of A, 
b,5, is a factor of the second row, 0,0, is a factor of the 
third row, and so on. Hence 


il 
3 
| 
_ 
— 
— 
~ 


: by be b, by bs eee Dy be sei 
i=0 1 Do bobs bo bs by eee bobs aN b 
1 bs bs by b, dy bs eee bb, oe vee 


1 b, b, On43 b,, Dasa On+9 ee b,, Bn+t a: Don 2 


APPLICATIONS AND SPECIAL FORMS. 151 


Skew Determinants, and Skew Symmetrical 
Determinants. 


119. We have heretofore shown (108) that the square of 
any determinant is a symmetrical determinant. If we now 
write the determinant of even order 


Phy Dy ty dy 1. SA Bh) — d, c, 4; 
| Osa Ue hao Oe —b, dg —ady 
Gyinby Cy lg —bz; dg —d3 Cs 
ei 76; Oy —by a -—-G & 


we get, by multiplying these factors together, 


A? = 
0 — (aby) — (ede) — (M03) — (Gd3) — (ad) — (Gd) |. 
(dbo) + (dy) 0 — (dbs) — (Cog) — (Goby) — (Cody) 
(a)b3) + (G3) (M23) + (Cody) 0 — (dgb,) — (C504) 
(Gh),) + (dy) (Gedy) + (Cody) — (gy) + (e504) 0 


In this determinant each element is equal to its conjugate 
with opposite sign, and the elements of the principal diagonal 
are zeros. Such determinants are called skew symmetrical. 
In other words, if in a determinant we have a,=—a, and 
a,,=0, the determinant is skew symmetrical. If a, is not 
zero, we have a skew determinant. It may be shown that the 
square of any determinant of even order can be expressed as 
a skew symmetrical determinant. Thus, since 


A={% Ae Ug Ay roe  Oqpn--8 n—2 Ain -1 On 
Clo; eg pg Ung °2* Cong Con—2 Con —1 Con 


On—-11 On-12 Un-13 On-14 °°° AOn—-in-3 An-1n—2 An-1n-1 U—in | 
0) 


ni Ong Ang On4 a Pek Onn—3 Onn-—2 Onn—1 Qon 


152 

=|tg —% AU 
Clog = — Ay, = On 
On—12 —An-11 An—-14 — %n-13 
Une an Ani Ong 


— 3 


== (15 


THEORY OF DETERMINANTS. 


eo Qing —Ajn-g Gin ——~Cineeee 
*** Qiong —Aon—3 Am —Aen-1 

48 Cl ee On-1 n—8 An—1n —QAn-1n-1 
ees Ann-2 — Ann-3 Onn Ann—1 


we have, after multiplying these determinants together, 


A?=| 0 mp 
My, O 
Ms, Meg 
Mn Mng 


For 


Myg 


Min ° 
Mon 
Men 


Jas 0 ny,=— Mays 


Mix = Ai Ug — Ajgyy + Aig Uns — Vig Aig + ose) + in—1 Vin — Vin Uin—15 


and hence 


Mi = 0, 


and 


My, — My; 


120. The consideration of skew determinants reduces to 
that of skew symmetrical determinants, as we shall now show. 


I. By 47, 
AM=|ay, yp 
Clo, Age 
nj Ano 


Ain | 
Con 


“ 


Ann | Win = — Ang 


= Ay + SC, Ag? + SC, Ag + ++» + 3C,_2Ag + C,. 


Now, since a,,—=— 4d, the determinants A)”, A)”, A)”, 
---, are all skew symmetrical, and A” is expressed in terms 
of skew symmetrical determinants. 


II. If, further, a, in A™ is equal to x, we have 
AM = Aj + 2A,» + 27 SA,-?) + Eek +- oA) 41a. 


It will soon be shown that a skew symmetrical determinant 


of odd order vanishes. 


Accordingly, the terms of this expan- 
sion in which the degree of A) is odd will vanish. 


Thus 


eer ee eS Ay keen AN Clee ae 
io, a » wick 7% 85 4 


APPLICATIONS AND SPECIAL FORMS. 153 
* —a —b —c|=|0 —a —b —-e | 
a «2 —d —e a 0 —d =e 
nie O Di ae f ad 0 —f 
eee 3 f x cece Uf 0 
e +2 0 —d —e|+/0 —b —c|+/0 —a —c|+!10 —a —b 
oe fee — St |b 0. —f\ 1a. 0° Hel lao ee 
a a Oye Fa 230) Gee. aot Bass 4 


Bee ssriiieittviisn 


a +a [0-+0 40] +2! 

FE, 

cS M4 ( (7 +274 04°40? +a’) w+ (af—be+cd)?’. 
The student may show that 


A=} a bo c¢ da |=(@?+0?+e?+d’)’. 


—b a —d c¢ 
—c da —b 
—d -—c b a 


Writing another skew determinant A,, whose elements are 


 e, f, g, h, in the same form as A just written, we see that z 2 
ASH (P+ PP +9? +h’)’. If we multiply A and A, together . 
_ by rows, we get another skew determinant A,, of the same . 
4 form as A and A,; the value of A, may accordingly be written 
a (m? +n? + 0? + p’)’, 
= where | 
-_ m=ae+bf+ceqg+dh, 0 =—ag+bh+ce —df, 
n=—af+be—ch+dg, p=—ah—bg+ecft+de. 
We have then . 
a ee 
BAS (CHP HCH AY (CLP GP HWY = (m? + 12+ 0 + py’, ies 


Dor HELE H A) OFPtG +H) = (mi tn +o +p), 


_ which is Euler’s theorem. 


154 THEORY OF DETERMINANTS. 


2 
121. Returning now to the consideration of skew symmet- 


rical determinants, let us take the two minors M and M, of 
109, and making a,=—a,;, 4, =0, the determinant itself is 
skew symmetrical; M becomes MM’, and M, becomes M/. 
Now since every element of M' equals each element of 1! 
with contrary sign, or since 


/ ! eee eee ! = —_— _ eee 
MM! =| Oy, An Ay , and M!= Cgp ggasi 
gg on Ag; ee — Ang — ng 

Gy —Qg 


M' =(—1)"My', 
where m, as before, is the order of the minors, 7.e., the con- 
jugate minors of a skew symmetrical determinant are equal if 
m ts even ; but if m is odd, the conjugate minors are equal, with 
contrary signs. 
In particular, if n is‘odd, A,,= A,,. 


But if n is even, Aye ela 


122. If the skew symmetrical determinant 


A — 0 yp Cg 
— pe 0 Ags 
—g —Ag JO 


is multiplied by (—1)*, we obtain 


—-A=| 0 —Q. —Aahg |. 
C9 0 Sr Co 
Cys Clog 0 


But since the rows of A are the columns of —A, 
A=— rie or A=: 


It is obvious that, in general, the effect of multiplying a 
skew symmetrical determinant A of order n by (—1)” is te 
change the rows into columns. Hence, when n is odd, 


Ao 


APPLICATIONS AND SPECIAL FORMS. 155 


Therefore a skew symmetrical determinant of odd order 
vanishes. In askew symmetrical determinant A,, is, of course, 
skew symmetrical ; hence 


123. From 121, where n is odd, the reciprocal determinant 
is symmetrical ; and, if n is even, the reciprocal determinant is 
skew symmetrical. 


124. I. Consider the following determinant 


A=|0O0 -—G. —Qyg —y |5 
- Cy 0 —s3, gg 
3 Ang 0 — Asa 
Cy, ng sy 0 


and the reciprocal determinant 


— Aj; — Az, 0 As, 
=A - Ay Ay 9 


Now, by 61, 


0 Ay, = A * 0 — Apog \ 
—A,, (0) Cog 0 
A, 3 
as 
Sie Ai? = fared ae or ra 29 
Cys 


and hence A is a perfect square. 


II. We shall now show that, in general, a skew symmetrical 
determinant of even order is a perfect square. 


Let 
9 Vie 0 Ae ** An Any *e* Gyon-1 =n 
(o1 0 -*° don Agn+t1 -** Agon-1 gan 
Ani An 2 ve 0 On n+1 se On on-1 An on 
Anti An412 °°° Ansin 0 -°* Ona an—-1 An+1 an 
Qon—11 (on-19°°° Aen—1n Con pe Das 0) (on—1 Qn 


Cont ~Qoneg °** Conn onntt °° Gon on-1 0 | ip — Ape 


156 THEORY OF DETERMINANTS. 


Then, as above, 


Aay Aa on | A: Aa, Gangne (a) 


Dap, 1 Daon Qn 


Now since A is skew symmetrical, and 7 is even, 


Day a Daonsn as 0 } and Dajon ee Dani 
te 
F 2 S| 3 = 2nl_, b 
“A @2n1 — A: Aa, Bongn OF A= A... ( ) 
G11 F2n2In 


Therefore A is a perfect square if Ag), anon 18 a perfect 
square. In other words, a skew symmetrical determinant of 
order 27” isa perfect square if one of the next lower even order is. 
But it is obvious that a skew symmetrical determinant of the 
second order is a perfect square, and we have shown above 
in I. that one of the fourth order is a perfect square; hence, 
by what we have just proved, a skew symmetrical determinant 
of the sixth order is a perfect square, and so on. Hence the 
theorem is true universally. 

For a simple illustration, let us apply (0) to the following 
determinant : 


A=/0 -—a@ -y -—z|=/—w —-y —2z/? 
xe’ O —-t —u 0 -t —wUu 
he 0 —-—v t 0 —v 
pee A 0 = (ve — wy + tz)’. 
| QO —t 
<>) 


As another application, we establish the following relation : 


9 (de— As)” (;—4)? (Ag—ay)” (Ag— Ay)? (A; —Ay)? (3 — Ay)? 
= [ (G.—a3)° (a, —a4)?+ (d3—,)’ (,—A4)? + (a —ay)® (ds—a4)°]?. 


The first expression equals (see example 7, page 37) 


a: .af a 1x ]1 —8o; San 
Cs et eee 1:-—3a, 3a, —a, 
Cgc, ate week 1 —38a,; 8a,” —a,’ 
Of 0 aoqee 1 —3a, 30/7. eee 


APPLICATIONS AND SPECIAL FORMS. 157 


= 0 (dy — a2)* (a, — as)? (a= a4)? 
(a, — a,)° ) (dy — ag)? (dg — Oy)® 
(a3 — a)° (a3 — d2)* 0 (a3 — (4)° 
(@—)° (a,—a,)® (a,—a,)® 0 
=|(a,— 4)? (a,;—a3)? (a,—a,)3/? + (dz — as)° 
0) (dg— 3)? (dy — 4)? 
(ds — dy)° 0 (ds — a,)° 


— [ (a— a3)° (a4 —a4)? + (a3— Ay) (yg — Ag)? + (ay —y)* (dg —ay)* |’, 
as was to be shown. 


125. The following proof of the preceding theorem has some 
advantages over the one just given. Let A be a skew symmet- 
rical determinant of even order. ‘Then Ag,, vanishes. Let 
By, be the complementary minor of a, in Ag,,, and hence a 
second minor of A. By 60, 


Bu Ba |= 9; 
Bi Bux O) 
and since mae Cae = pir - (2) 


Expanding A by Cauchy’s theorem, 63, III., in terms of the 
elements of the first row and first column, we have, since 


Bay — 0, 
A =— 34; Oy Ba = Udy V BiB, substituting from (2), (3) 


in which i, k have the values 2, 3, --- 2n. From (8) we have 
at once Ds 
A = [2a V Bul’. 

Here A is expressed as the square of a linear function of 
the elements of the first row. This function is rational if 
VB; is rational. But B; is a skew symmetrical determinant of 
order 2n —2. Hence a skew symmetrical determinant of order 
2n is a perfect square if one of order 2n—2 is. But we 
proved (124, I.) that a skew symmetrical determinant of the 


158 THEORY OF DETERMINANTS. 


fourth order is a perfect square; hence, by what we have just 
proved, one of the sixth order is a perfect square, and so on. 


126. Since 
A = [2a V Bil? 
= [dy V Bop + Ay3 V Bs3 + Oy VER + +++ + Oo, \V Barua ; 


that is, since A is the square of a linear function of the ele- . 
ments of the first row, we see that if A is of the fourth order, 
VA contains 3 terms; then, if A is of the sixth order, V A 
contains 5.3 terms, etc. In general, then, VA is the sum of 


(2n —1) (2n —38) «-- 5. 3. 1 terms. 


Every term of VA is, moreover, the product of n elements of 
A, in which no subscript is repeated. For, taking the term 
dina eae for instance, we see that it consists of terms in which 
neither of the subscripts 1,4 is repeated. But By will contain 
a term 3/733, in which, as before, +/733 contains none of the 
subscripts 1,2,3,4; and so on. Hence VA is the sum of 
terms of the form 
Ay2 Ug4 Usg *** Aon—1905 


in which no subscript is repeated. 
If A is of the fourth order, for example, we have 


4) 
AM%=| 0 dy yg Ay], and VA® = (ayy day + Aig £ Ay, Ans). 
Clay O Algg Cog 
Az31 Ago O Cay 
Ag, Ay yg O | Ay = — Ay; 


To determine which sign is prefixed to each term, we observe 
that since the interchange of two subscripts of A amounts to 
an interchange of two rows, and also of two columns, and 
therefore leaves A unchanged, /A must be a function in which 
the interchange of two subscripts either causes no change or 
simply a change in sign. 


Sa 


APPLICATIONS AND SPECIAL FORMS. 159 


If we consider any term of VA®, as dy.a4, which the inter- 
ehange of the subscripts 1,2 transforms into Cg, Og, == — Ay As4y 
it is obvious that VA does change sign on interchanging two 
subscripts. We have then the square root equal to 


Ay2 Ag, — Aqg Ung + Ay4 Mos 5 (2) 


_ for, if the second term of (2) were +, the interchange of 2 


and 3, while changing the sign of the last term, leaves the 


signs of the first two unchanged. 


Since @,—=— d,, it is always possible to so interchange the 
subscripts that all the terms shall be positive. Thus 


(a), C2 
VA a Ayo Ass + Ayg Ago + Ayy Ao3. 


127. In general, we proceed as follows: 
A being a skew symmetrical determinant of the 2nth order, 
A contains the term 


9 
( —1 ) ™ Azo Ag Ags Cbg Cbsg Ogs *** Clon—19n Cen 2n—1 = (dys Az4 sg °** Aon on) ve 
Hence VA contains the term 
F Aye Mg4 Asg *** Agn-19, = TI. 


The positive square root of A which contains 7 as its first 
term is an important function, possessing many properties 
analogous to the properties of determinants, and is called a 
Pfaffian. The notation 


Belle, 2s ore (ly 2 rages 2%). 


has been adopted for the Pfaffian. From what precedes, we 
see that the terms of the Pfaffian are obtained from the prin- 
cipal term by permuting the subscripts 2, 3, --- 2m in all pos- 
sible ways, and changing sign with every permutation. 

Since d;,=— Q,, we may so arrange the elements that every 
term of P is positive. Thus in the case of A® above we have 


VAY = P= Ay digg + Ag dye + yy Mop. (p) 


160 THEORY OF DETERMINANTS. 


128. If two subscripts are interchanged, the sign of P is 
changed. Let a,,8 be the terms of P containing the element 
d,,. Then the elements of 8 do not involve the subscripts 
rand s. Interchanging 7 and s, let P become P'’. Now 


P? ad Hoe 


since each square is A, in which two rows and also two columns 
have been interchanged ; 


ere fe —— aif P'.~. 
But because of the interchange in 7 and s, 
d,, 8 becomes — 4,8 5 


or, since-the term a,,8 of P=—a,,8 of P', it follows that 
P=— P', as was to be shown. 


129. We shall now prove a theorem by which we may com- 
pute Pfaffians of order 27 from those of order 2n — 2.* 
Assuming 


WV Bi = (—1)' (2, 8, --, 7-1, (41, +, 2n), (1) 
or, after making ¢—2 cyclical interchanges, 

VBg=(i+1, 742, +, Qn, 2, 8, +, C27), (2) 
where 8 has the same meaning as in 125, we show that 

V Bi V Bix = Bx; (3) 
and then since 


Po lag V Bos + Cys V Bes fees + Oho V Biss (4) 


* There is a difference in the nomenclature. We have here considered 
the order of the Pfaffian to be determined by the number of subscripts 
involved. Some authors determine the order of the Pfaffian by the order 
of the terms in the elements. Thus (1, 2, 3, 4), or ||a,,|, which we have 
designated as a Pfaffian of the fourth order, is said by some writers to be 
of the second order. 


2a APPLICATIONS AND SPECIAL FORMS. 


2,3, + es ae soy 21) + dy (Ay + AE) 
a Fins (2,3 aes ,2n—1) 


Ss 


To show that upon the assumption (1) or (2) the equation 
ac) results, we proceed as follows: 


Since Bis Bu a Bes 


- the terms of VB; V8, must be equal each to each to the terms 
of Bx. Or equal with contrary sign. The product 


Paty (2 Bice tet 1, 2 2) 
(2; 85 3-) B—1, k+1, ot 


becomes, after a certain number of interchanges, 


(A, Py 51% 0085 Us v) (Ds Jy M5 Sy erty Us ty; (7) 
; es where Ps, 75 +++; U, v denote the series of numbers 


2, 3, aa 2n, © 
exclusive of i,k. Again, 


i id eee 
Ba=(—1)** ge eg °° Agz-i Oe n41 
Ggq Ugg (ces Agz-1 Ugi41 °°° 


eee eee . ‘ (8) 
Opie 3%7235° 722 Oe ea ei een * 
Misia Gs1g °° Ginter Wisin 


(Grom = 0, ns rep etee Mts, ) 


z becomes, after the same number of interchanges as were em- 
ployed to change (6) to (7), 


Any Ang An °** Ayy Any |- 


App AUpqg AUpr *** Apy Api 
Dap gq Agr *** Ag Agi (9) 


| Ayp Ang Uy 98 Ayy Ang 


162 THEORY OF DETERMINANTS. 


Now the first term of the product of (7) is 
Vp Ugr *** Ayy Mpg Uys *** Cig - 


which is identical with the first term of the determinant (9). 
Whence the truth of (2) is established, and (5) gives the 
desired expansion of P. It is to be noted that the successive 
terms of P are written cyclically. For example, A® being a 
skew symmetrical determinant of the fourth order, 


AWS" = (152,354)-5 
and (1, 2, 8, 4) = Gy dgg + yg Ugg + yg Aog- 
AO aa PP (15 2, 8 fest he 


(1, 2, +++ 6) = dy, (3, 4, 5,6) + ous (4,5, 6, 2) + dy (5, 6s 2, 8) 
+ ys (6, 2,3, 4) + Aig (2, 3,4, 5) 


= Ayo Ag Use  Ay2 Ags Ugg + C12 Ugg Ogs 
H+ yg Ags Ugg + Aq3 gg Mos + Ayg Cas Ase 
+ Ay4 Agg og + Ay4 As2 Ugg + Aqy Ags Ugg 
H+ Ay5 Ugg Azq As Ags Ayo + As Ags Clos 
HF yg yg Ags TF yg Mog Ugg -F Ag Ags Ag4. 


130. The student must have already noticed the analogy 
between determinants and Pfaffians referred to above. The 
following notation, based upon this analogy, is interesting. 
Since the Pfaffian involves just half the elements of a skew 
symmetrical determinant like A of 124, II., we write the 
Pfaffian 

P=lay. Ms Ay +** Gia Gin |s 
Gog Ung *** Agon-1 Agen 
Usa °** Ugon—-1  Agon 


Con—22n-1 Aon-29n 


, Oon—19n 
which is shortened to 


|| yy Clog Ogg +++ Gen-1on|, Or to ff(dion), or to || 1 on . 


APPLICATIONS AND SPECIAL FORMS. 163 


In particular, we have for a Pfaffian of the third order 


la, bd) G = IP(h b2¢3) = ||a, by Cs |. 
bs G 
C3 


We may accordingly write equation (p), at the end of 127, 
VA® = I[@4], or rather A® = |/a,,/?; 
and the general equation would be 
7a ia — 1 Aa pe 


131. We must here conclude the discussion of Pfaffians with 
the theorem: a bordered* skew symmetrical determinant is 
the product of two P faffians. 

From equation (0), 122, II., 


Maen = A+ Aan, donon* 
+ May, = (1, 2, +++, 20) (2, 8, +, 22 —2, 2n—1), (1) 
which proves the theorem when the determinant is of odd order. 
Let A™ be a skew symmetrical determinant of odd order. Ag,, 
is a skew symmetrical determinant of even order, and hence 
eee (18 (1, 25% y-, 21, tf 1a) .n) 
= (i+1, +--+, ”, 1, 2, ---,7—1). 
Now A” being zero, we have, by 60, 
Mai = 4a 
°. Ag, = (+1, --, n, 1, 2, +++, 7—1) (2) 
(K+ 1, +5”, 1,2, ---, k—1), 
which proves that a bordered skew symmetrical determinant of 
even order is the product of two Pfaffians: for any minor Aq,, 


* A bordered skew symmetrical determinant is one in which the minor 
of one of the corner elements is skew symmetrical. 


164. THEORY OF DETERMINANTS. 


of a skew symmetrical determinant is evidently expressible as 
a bordered skew symmetrical determinant. 


If 
ad... = — a 
A=/dy| ” *", we find by (1), 
ay = 
Dag =—|%2 Gg Ay Ay he 
Clog ag) ng §=— 5, ag 
Upp ay Ug, a 
Msg C4g gg gs Cys 
Qso. Ass sy Css sg | 1, =-0,4,, = — O,, 
=—(, 2, 3,4, 5, 6) (2, 3, 4, 5). 
Again, if 
C—O A 
A =| os eas *", we find by (2) 
by = 
Ray = |G og Ags Cg 


Cs, Msg sg sy | Aig =O, Ag = — sy 


as the student can readily verify. 


Circulants. 
132. The resultant of 
J(#) =H? +a,%+a,=0, (1) 
by Sylvester’s method (92) is 

% @ as O 0 |=|a, @ O|=F. 
Dove ore Oats am OY As Qi Qs 
Ot tee ah Td og On “Oy is 
1 Oe emer Ler) 

Bi Demet ee eth yo aE 


ij 
‘ 
‘ 
‘ 
J 
: 


APPLICATIONS AND SPECIAL FORMS. 165 


Now a, a, a3 being the three roots of unity, it is evident 


(94) that 
R = f(a) f (a2) f (as) ; (3) 


or, denoting one of the imaginary cube roots of unity by a, 
the other is o”, and we may write 


B= fA) f(a) f(a) 
= (0, +4. + ds) (dy0? + da + 3) (ya + doa? + ag), 
an equation exhibiting the factors of R. 


133. J? is evidently a symmetrical determinant formed from 
the elements a, d,, @3 in its first row, in such a way that the 
last element in every row is the first element in every succeed- 
ing row, and the other elements are written in order. Such a 
determinant is called a Circulant.* The intimate connection 
of the Circulant of the third order with the cube roots of unity 
was shown in the last article. We shall now prove that, in 
general, the circulant of the nth order, 


C=C (a,a,+--a,) =| a, G Gg > Any 
Un Oy Ag ike Oe a acs 
Gn-1 Gy, Ay °** Ay_ 3 An-2 
As Os ds et Ay A» 
Certs le An ly 


is the product of all factors of the form 
Ay, 04" * + yy," + My ga” ® + +++ +307 + doa, +) = F(a); 


in which a; is one of the nth roots of unity, and ¢ accordingly 
takes successively all the values 1, 2,---. In symbols, we 
are to show 


C' (a, dg Mg++ Ay) = LF (a,a0" + Gy _10,;"-? + +++ + Aga; + a) 
= f(m) (a2) f(a3) +++ f(an)- 


Write another determinant of the nth order 


* The Circulant is of frequent occurrence in the Theory of numbers. 


166 THEORY OF DETERMINANTS, 


A — i ay a; rid Ts Oy 
1 %Gge eae ag rae 


Parga age Woe dee 


2 -l 
1 An An da ie ai Cat 


Multiplying by rows, 

CA=| f(a) Toy)? 2 fGeee F (on) 
ar f(a1) a2 f(a2) an iF (Gn) On (an) 
ay f(r) az f(z) a nat oe 1) a, f(4n) 
Cag) (0) ay *f (a2) Wad 1 f(an-1) an" f(an) 

Factoring this product, 


CA= I (a1) f(a2) ai F(an)A- 
CO =f (a1) f(a) +++ F(an) 


1 
-1 —2 
= I (4n ar + Oya, +++ + Aga, + ay) 
‘= 


For an illustration, |7 0 0 O y 
5a 
0 arias ae Oa 
OF 00g sare 
Oa00 0S 


= (a+ ay) (+ any) (@ + ogy) (@ + ay) (@ + a5) 


=@+N(2+| vent, ¥i02 ae 
(«+[ vies V0 =i|v) 
(ap one Va=B y—j)y) 
(=+| - ee V10—2-V5 vio 2a =i|n) 


=O + y, 


as was evident from the beginning. 


APPLICATIONS AND SPECIAL FORMS. 167 


134. The circulant of the fourth order 


Gs 10) dy yO, 
Qa Oy enOs Oe 
gs oer Palues 
Oe Be OL cy 


can be expressed as a circulant of the second order, as follows. 
We have 


Bete) | Oy — ie Og  — Oy, | =| Ga ly) gy} 
Gg —Q- % —Q, lee ee Oy at: 
Q4 —Q, GW —AOs aed. Uy, a, 
do —g OQ, —Qy, aR Oar 


The first of these determinants is obtained by interchanging 
the second and third rows, and multiplying by (—1)?; the 
second is obtained from the first by reversing the order of the 
rows, and then reversing the order of the columns. 

Multiplying them together, 


C= 
7 —2 deta? 2a,a,—a°—a/ ) 0 : 
2g — Ay — Ay As’ —2Ayao tay OO 0 

0 0 2 Ase —,’—a,7 a +a’—2 ad, 

0 0 Age —2Adg tae 2 dey —As?— Ay” 


Whence expressing C’ as the product of two minors, and 
extracting the square root, 


O = | Ay — Ag dy + Agdz3 — Ag, Ag Ay — Mg Ag + Ay Ag — Ay 4], 
Az Ay — Aga + Ay, Ag — AyAg Ay Ay — Ag Ag + Ag Ag — An My 


as was to be shown. 

The method employed in this special case is equally appli- 
cable to show that, in general, a circulant of order 2n can be 
expressed as a circulant of the nth order. 

We add the following proot, bowever, which is based upon 
the fundamental property of circulants. 


168 THEORY OF DETERMINANTS. 


We have to show that 


C=| a, Mg Ag *** Agn_1 Aon = | 0, 0g. +* DO nmeee 
Con Gy Ag *** Ang Aen-1 By, By +8* Ong Ona 
Gon -1 Aon A, *** *** Ame G24 b,, et Dare b,~2 (1) 
As 4 Cs pean Cy Og bs by Chet! Dd, by 
As Og 4 pies) Con Cy by b; oe b, Dy 
where 
By = AQ, — — Agn Ag = Ayn_1 3 — + *** — Ap on 
by — As Ay — Ag Ag + A, A a eof't 4 — Ay Con 
bs = As, — Ag Ag =f Ag, —t ees — Agon 


Dy = Myp_1 hy, — Agy_9 Me + A, _g 3 — + +++ — Ag lon. 


The first determinant 


i=2n 


al Qn—2 on— 
C11 (Con aj” + Can 1p”? Ayn _gj? vee Ay a;+ dy). (2) 
i=1 


Now for every 2th root a of unity there is one —a. Hence 
(2) may be written 


C= TL b,ae** ita Dna agents Sr aE b3a;" + be a,” = by) e (3) 
If +a, +o, tos, tay +, ta, 
are the 2mth roots of unity, it is evident that 
ey, gee ag weeks Cs 


are the nth roots of unity. Hence the second member of (3) 


equals the second determinant of (1), which establishes the 
theorem. 


For example, 


Cz=\a 6. c d|=) 29 re 
GEE DC Beno. 
CHO Op 
ER hae Ate 


APPLICATIONS AND SPECIAL FORMS. 169 


in which 
H=d+c¢ —2bd, F=—b?— + 2a. 


“. C= (a? +c —2bd)?— (2ac— 0? — a)? 


Centro-symmetric Determinants. 


135. If we suppose a determinant to be symmetrical with 
respect to the centre of the square (centro-symmetric*), we 
have, if the determinant is of order 2n, 


A=} a) A 92 Cai Gn Oy Oy 29° On Pe Uae {= 
Co, se °° Gon-1 Aon 0, by ¢© Oon-1 Dan 


Ani Ang *°° Ann-1 Ann Oni Ons eo” b,, n—1 Dee 
Onn b,, es rh b,, 2 Bra Onn AUnn-1 °** Ang Un 


Bon Don *** Doe be Gen Men-1 *** Age Ay} 
Bin Oyn-1 *** Oye Oy, On Un-1 ** Gq Ay 


We will transform A as follows: add the last column to the 
first, the (2n —1)th to the second, and so on, finally adding 
the .(n+1)th to the nth. Afterward subtract the first row 
from the last, the second from the (2n—1)th, and so on, 
finally subtracting the nth from the (n+1)th. Then 


A=] dy + Om ++! Gin + Oy Dy =e Din 
Gey + Don *** Con + Oo bx aes Don 
Ani + Don -** Ann + Dry bn Riek Dan 

0 a 0 Onn — bn wn 5 Gan 

0 ae 0 Can — Do, +++ Ao — Dan 

O eee 0 Ain — Ds spas Ay aoe Dy, 


* It may be shown that the product of any two determinants of the nth 
order is expressible as a centro-symmetric determinant of the 2 nth order. 


170 THEORY OF DETERMINANTS. 
Hence 
A= a, +0, GM t+ Bin-a °** Cina + By an + by 
ln + Don Gea + Dona *** Gana +902 en + On 
Any + One Ong + Den 3 tee 9 Cy + Dig Onn + Ba 
» 88) EE ere Day Onn—1 — Dg cee Ong Dini Ani — Dan . 
Con — 09, Gon — Dog + Cag — Don 1 Ag — Don 
Oy, — Oy An — Diy 88 Cyg — Ow Gu Din 
Tf A is of order 2n +1, we write 
A= Ap Clin ky by : Oty ee 
Gg, Are Cy, hy by, - by, ea 
| An) Ong Onn k,, bia A Dos By 
ma ly L,, fot. vee dy l 
| 
/ Dan Re bd Diy k,, Onn °° Ang Cn 
| Bin Ona sts) On ky Gin Aye Ay 


By making just the 


find 


Ajo + Dini 
Gee + Don 1 


Ano ae OneT 
21, 


Ann-1 = Dig 
Qon—-1 — Doo 
Ain-y — Oyp 


same transformations 


Cin + dy 
Chon + bay 


as before, we 


ky 
Ky 


aa 


APPLICATIONS AND SPECIAL FORMS. 171 


Collecting results, we have: a centro-symmetric determinant 
equals the product of two determinants each of the nth order, 
if the order of the symmetric determinant is 2n; if the order 
of the symmetric determinant is 2n +1, the factors are of order 


n and n+1 respectively. 


For an illustration we expand the following determinant : 


Liew bora g 
Parl efor i O70 
fee, RP OS Oo 
ie & patton 
ee es ie G 


Lag 


h 
e 


=|jath b+g c+f dt+e|x|a—h 
b+g ath d+e c+f b—g 
ec+f d+e ath b+g c—f 
d+e c+tf b+qg ath d—e 


h 


oe 


ath+td+te er < Phils 


b+g+tet+f ath+d+e b+g—c-—f 

x| a—h+d—e b—g+c—f|x|a—h—d+e 

b—g+cec—f a—h+d—e b—g-—cet+f 
Continuants. 


c—f d—e 
d—e c—f 
a—h b—g 
b—g a-—h 
b+g—c—f 
ath—d—e 
b—g—c+f\. 
a—h—d-+e 


136. Consider the three simultaneous equations : 


(a) 8a,— 2% 


==y1 
(0) & + 42% — ae 
(c) Xo +52, = 0 


1 We THEORY OF DETERMINANTS. 


From (@), 
®\ 1 
a( —2)=1; AO So DE = 
Uy 
From (0), 
Lo 1 1 
a ay 7 [ma 
1 3 
%5 Ar __ 
Xo 
From (c), 
hl ones ee Pp 
Ave 
Fa; 


The value of x, is thus expressed as a continued fraction. 
If we solve for 2, by 69, we find 


Getta: beat i Oealee ee een . 
(jit Aa emasy i ae 1 
Cs itl (ee aes 


We see then that a continued fraction may be expressed as 
the quotient of two determinants. 
We shall now proceed to the application of determinants to 


continued fractions in general. 


137. From the simultaneous equations 


(1) Ay X —e Xo — ay 
(2) apa + pM, = ay 
I -(3) Og Xo + As U3 = Wy 
(n— 1) On -1X_n-9 + Ay_1 Xy_1 = Ly 


(n) ay Ly —l ata Ay, x, == Cn+1 


APPLICATIONS AND SPECIAL FORMS. 17s 


we obtain from (1) 


as obtained from (2), (3), --- (~—1), (), we have 


ay 
1 a Fag 
Ag+ *, 
ne ee 
An-1 + Gy 


The value of x, is seen to be expressible as a continued 
fraction. If we stop at the nth quotient, and thus take the 
mth convergent for the value of a, then a,,, and all the suc- 
ceeding xs must be conceived to vanish. In that case 2, is 


the continued fraction. 


fr Se ay ag a3 An-1 An 


~ , +g +. +a, +a, 


The consecutive convergents to & will be denoted by 


fete 
The determinant expression for —* is now found by making 


n 


41 = 0 in equations I., and solving for x, by 69. We find 


174 THEORY OF DETERMINANTS. 


oy — 1. eee 0 °0=ane 
0 a —l VU 0 0 0 0 
0-5 ag eaewenetgreO 0° 20a 
O° COU eee 0. 3:05am 
027 ,-0 Soe OMe) nee PA a 
Oy. 20" SF Oe 0 tag ene 
“= ; 
ry Meepe Eas emai 85: 0 > Sue 
dete ey) ae 0 0 0 0 
Ciutat — lie 0.7 “Osama 
0. 0. 0.0 - 0...) Ga 


050° 0 0. 0 \.0 Ce 


which is the determinant expression sought, and hence is 


Jes 
Q, 
Looking at numerator and denominator of this convergent, 
we see that 
i BOLT, IQn 
da, 
and thus dQ, 
a; —— 
F= oh and .. f= a, © M08 An), 
n day, 


138. A determinant having the form of Q, in the preceding 
article is called a continuant ; 7.e., a continuant is a determi- 
nant in which the elements outside of the principal diagonal 
and the two adjacent minor diagonals are all zeros, and one of 
these minor diagonals has each of its elements —1. 

Since 


Oy A 0 0 0 0 0 
Qo Ag —— 1 0 0 0 0 
0 a3 As —] 0 0 0 
0 0 0 0 ees Un —1 ae 
0 0: acini 


APPLICATIONS AND SPECIAL FORMS. 175 


eT Oy ene tienes ¢ We si: 
—1- As as 0 oes 0 0 0 


0 —1 As a4 coe 0 0 0 
0 0 Pees. 6) eae 
0 0 ier Oo eae. OL alee 


n 


it is immaterial on which side of the principal diagonal we 
write that minor diagonal whose elements are —1. Also we 
may write 


eee ia. (a) oO 0 eC 0 0 
ag Ao —1l 0 0 0 0 
(aera Gs) 1 0 0 0 
0 0 0 0 a ok 
ea) ist) 0 Ae ads 

Peet. 0. 0 0 0 Geo 
= Oo Ag 1 0 0 0 0 0 
0 — Gs As 1 0 0 0 0 
ee Nai a ee 
(He eee rete: Oe. a... a: 


We shall employ the following notation : 


Q aa Ag Ag eee On 
- (Ly Clg Ag *** Ay 


Returning to Pn we may now write 


n 


176 THEORY OF DETERMINANTS. 


n em) ag Eig 8 2 
Cy Clg ah hd An 


or, the nth convergent to a continued fraction F is expressible as 
the quotient of two continuants multiplied by the first numerator 
of F. 


139. Expanding P,, in terms of the elements of the last row, 


we find Pais 
if = ay 3 4 de 
Oy 


pages eile ye) 0) Be ae! nH 0 0 


ieee. 1-80. A is 
0 4 A, —1 CO 0) 0 0 


0 0 0 0 Oo On eae 
6 a0 0 0: alae 
_ a, A) Cy <7 1 0 ) 0 0 0 
Os Cs —l 0 0 0 0 
Os 6, eet I maa ,. es 
0 0 0 0 G,4 Opa — 
OR 0 Rae 0° cag Game 


oa ag 4 eee a mas | ag O4 eee a set 
r= An ay ( ) as A, Ay ( eb ) 
hg hg 2 ey Gg Ag se" | Os 


— An Feta? “te an betas (A) 


Similarly, 


0, = Ag Og eee An a Ao Og eee a | 
n 
Gy Ug, a hy Oy, Ag 89 Cay 


= on ( se date Res An_2 ) 
A Cg ay Oy _2 


= a, Qn—1 os me an Qn—2 (B) 


APPLICATIONS AND SPECIAL FORMS. lit 


140. It is to be observed that the equations (A) and (B), 
besides establishing the law of formation of the consecutive 
convergents to /, give the expansion of a continuant of order 
n in terms of continuants of lower orders. Thus, by (3); 


( a9 as 4 ) ( Oo ag a9 
ae a 5) 
Gy My Ag Ay SiN Gi Oy, hy phe a4 Oh, ae 


Qe : Og 
= Ag A4 dy Mp + dyaz(,) + a4 eng 
1 


= Cy Ag Az Uy + An Ag Ay + 1, A3 4 + Ay Ag 4 + Og Ag. 


141. Equation (B) is, in fact, a special case of the more 
general theorem 


Bee Oy 2°" Oy Az Gg 9% Ge Ap+2 °** On 
meg 8 ef) NO lg |) NO es 
Gg *°° Ay] Greg °** Og . \ 
+ Oy41 if 
Cy eee Oey O,+49 eee An 


This is easily proved by writing out the continuant of the 
first member in full, and expanding by Laplace’s Theorem 
(55) in terms of the minors formed from the first 7 rows and 
their complementaries. 

We may also use (B) to obtain another expansion of Q,. 
Thus 


Gp ag °** Ay s ag G4, °** Ay toy A, GAs *** An ; 
——_ § 9 
ads. se" =. C,, ON Ge Un eee "Nevis a henna (he 


as the student may easily verify by expanding the first member 
in terms of the elements of the first column. 


142. It will afford the student an excellent exercise to take 


the quotient 
a eee a 
ay : :) 
Gg Ag *** An 


Op ndg orm Cy ) 
Ay Co 5° (Bn 


178 THEORY OF DETERMINANTS. 


and, with the help of (C) of the preceding article, transform 


it into the continued fraction 


Q) a9 ag ay 


a, +4, +0, +0, 


143. 57 or 62 established the theorem 
(ene tas Oe dA dA (1) 


ae fe SSS 


day,da,, dd, da, dd, da, 


Let A=Q,= ( aes sd ) 


Cy, Oy ++ 

and let pectic ere Loe) aoe C nae 
Then GAS fag} dg VS 
day Ann Ag Ag se An-1 ‘a a ; 


where P,,_; has the meaning assigned to it from the beginning. 


Also GA fog xatag nies 
day, Clg Ag *** Ay ay 
Similarly, 
d dA 
= Say +++ Oy} = —— = (—1)" 
ayn Ady) 
and dA pe Ao Ag eee 3 | Ey : 
d Aa = Q,-1 b] 
8 Dees Ay Og eee An-] 
A ig Og Wert shag ag Ag °° An} 
Ay As eee An Os As eee An 
ag An a2 An] 
=( )( )-(y "ay ag On 
Ag Ag An A, As An- 


or Os Park rar se A a (—1)” ay agag-++ an. 


APPLICATIONS AND SPECIAL FORMS. 179 


144. With the help of determinants we may now show 
that 
Pac) aids Mele | OQ, O41 ay, 


Dt ts +O +o, +o, foggy se aoe 


equals 


F' —_ ef Qo As 1 a Vy, Oy Qa 


A, +, +4, + HO, 3 +O, + O41 + a, 


Express # as the quotient of two determinants, employing 
the form obtained in 137. Now transform numerator and 
denominator of this determinant as follows. Multiply the 
(kK —1)th column, and divide the (k — 1)th row of each deter- 
minant by —o,_,. Then perform the same operations upon 
each succeeding row and column. Afterward multiply the 
(k—1)th row of each determinant by x. The value of the 
fraction is not changed, and we obtain for the value of F 


Me ea ee OO 0.. 08-0: 0-0 0 
ee — 1.0 «0 Tallest 09> 2-9 0 
eee 00 0. COCO Oe OOO 


on) 
= 


a) 
ie 
co) 


eee), Os yg ayy «=O 0 


0 0 0 O «+ —U% a, 14 —a% O Je <see 20 0 
0 0 0 BS et ts eos ee Ba W 0 0 
0 0 0 Qe ete Q 0 0 0 0 Can eh 
ad, —1l 0 Ota. aU 0 0 0 O «+ O 0 | 
eed, —1, O «+ 0 0 0 0 0 0 0 

| 0 Gat. Oe — 1 ee 0 0 0 0 0 0 
0 0 0 ) os Oy_9 Op} 0 0 0 eee 0 0 
0 0 0 QO ++ —B@ 4x —a 0 O +. O 0 | 

| 0 0 0 O eects OF eee es One cine 1! 5: Oe 0 0 | 
0 0 0 0 0 0 0 0 0 Oy ink, Ob 


180 THEORY OF DETERMINANTS. 


Now divide the (k—1)th column, and multiply the (4 —1)th 
row of numerator and denominator by —a,_;. Then divide 
the kth column, and multiply the Ath row of numerator and 
denominator by —a,_,;#, and perform the same operations upon 
the succeeding rows and columns. We obtain 


la —l 0O QO «+ OQ 0 0 0. O «- O 0 | 
0 Ugo bay) Sete 0 0 Oipases 
0 ag Cs —1 ee 0 0 0 0 0 Lindl} ) 0 


ae 
So 
—) 


0 0 0 Da, 40 Oe > 1 0 QO «- O 0 
«0 (a2 a,  —P yee 
) 0 0 (i= Fees, G 0 On41 Any, —1 --- O 0 


oO 
=) 
om) 
j=) 
>) 
=) 


0-000 30°.. 0 06 +0) 0° ox 
@ —1 0:0 +0 0 .0 “0° /i0= nn 


ag) 7) dae — 40. ee 0 0 0 0 Q eee 
O ag Chg) ston) O O 0 O>. Sonar O 


—) 
= 


o) 0 0 0 5, 10 Aye —1 ) 0 #2 ) 0 
0 0 0 0 9s 00 Capa gee ee 0 
0 O° OO - Os 0 vege a1 


=>) 
co) 


10° OF O 0 + 0 (0+ 0°08 2 0 en 
But this quotient is the continued fraction F’. 


145. In a certain investigation it becomes necessary to 


show that the denominators D, and D, of the convergents 
to the fractions 


oO 


o b b 
Gy + A, a, 


By 
. ie 


P~) 
= lo 
+ 
=) 
RE 


are equal. 


We have 


D, =| a 


and 


APPLICATIONS AND SPECIAL FORMS. 


0: , Os ee 
Baa: 
Cae 
fo aan! 1 

Diets 

O27 9 

Or. 0 
U'GeD 
fy —1 

Vener 


181 


By reversing the order of the columns in D,, and also the 
order of the rows, and afterward making the rows the columns 
in order, the original determinant is unchanged either in sign 


or magnitude. 


to D,. 


146. The’ quotient 


can be expressed as a continued fraction, as 


| be Cs | 


Hine Og iCal 


1 
0 
0 


Cy 
Ce 
A3 


| by es 
bai Denks| 
follows: 

meni Vk OF || 
Os Cs 0a 0; Col 
een ds! Un. 10, Ca 

ee ; 
he eV aa 6,0 
ares Qe by 10,66] 
Da Ce lg 0, |b, Cl 


But by these transformations D, is changed 
Whence D, = D,, as was to be shown. 


182 THEORY OF DETERMINANTS. 


bs 0 0 bs 0 ~ 9 
(ly Ds | Dy | Dress | dy bg | b, —1 
1 0 bs Sere 0 — bz | Cs | | By cg | 
Old byl apne tense ge om —1 0 
‘| dab] by 1 bye | — b;| a2bs] bs i 
0 b; | dy cg | 0 — | b,c | | By eg | 
b, 
Pati phil ays 
bs Dz} b7 65) 


| Dy es | 
This process is equally applicable to show that, in general, 
the quotient of two determinants At ig expressible as a con- 
tinued fraction, provided only that 


Ras Sete ry AN GAs or A, = 292) ore 


a 1 ja : 


on day, iS Ayn tg Any dann 
147. The continued fraction 
Fos Se ee 
as a +a, +4, +d, 
is evidently equal to Oy Gg ** Gy 
My Us 55 
a2 a3 a at 
Cy As .e On 
For 
a, —1 0 O 0 O;+/|m—1 0 0.0 O 
0 a —1 0 0 0 0 a —1 0 =<. 
0 as As —1 . 0 0 0 Og Cg —1| bid 0 0 
Bare ct eee ae ee 0 O G, Gy rennuee 
010s 0380 Gy Gn| |ere see see Gen 


APPLICATIONS AND SPECIAL FORMS. 183 


But the first determinant in the numerator may be written 


0 —1 0 0 Ca Orta (yar ft =12 () 0 OSE Ats 
oo 1 (0 GecGre har ag) tae 0 0 
aver ds —1 -. 0 0 QO ag dy —1 =e eg 
0 0 ag Cs 0) 0 0 0 Ag Az 0 0 
00. 802 a, 0) Or GeO Gee ane 


whence the desired result is at once obtained by substituting 
in the numerator of the value of F,, and adding the deter- 
minants. 


148. We may, with the help of the preceding article, express 
the value of the periodic continued fraction 
i Fa +d, +4, Neen ntl 7, 
* * 


as the quotient of two determinants. (The * marks the re- 
curring period.) 
If we put x for the continued fraction, we have 


ad, +d, +43 ce +d +a, mM+2 


b, by it he bs Do by; ) 
M GQ, Mg s+ Gg A, (mM+2) 


HEF beady len oth Pak lk ) 
Oy Cp Cs Pa an) Cy (m+) 


clearing of fractions, and expanding, 


a be bs eee bs bo 7) ue 2( Do bs ie bs bo ) 
Cy Cg Lies Ag Ay m Cy (em) Ag Ay 


“pa b; by ta te by b ) ai x ( by be es be }: 
mM Ay Cp *** Ag Oy m m Ay 5, 2s As Ay 


184 THEORY OF DETERMINANTS. 


But the first term of the first member equals the second 
term of the second member of this equation. 


( D, be aii bs bo by z , 
m A Ad ses As Ap Ay m 
bo bs at a bs by ; 
Ay Ag As 22? As As Ay 
149. Let us now consider the ascending continued fraction 


Ag ee) a 
ane ® 2 0) oi Boe +o, 


As = 
Qo “fe Ay Ag 3 On 


PP, Pe. Br 
q Nn q2 An 


and let us obtain the determinant expression for the nth con- 
vergent. 
We have evidently 
Yn = Ay Ag M3 +++ Ay. 


Pn iS determined from the following equations, which the 
student can easily deduce : 


Pi = a) 
— AcP; + Po = de 
— A3P2 + Ps = a3 
—Gn_-2 Pn—3 + Pn—2 = ay_2 
— An Pn—s + Pn-1 —= Oy 4 


On Pn—-1 +. = a, 


a 


APPLICATIONS AND SPECIAL FORMS. 185 


From these equations 


Pr = 1 0 0 eee 0 0 @) ay 
‘ pita we BQ) eee 0 0 Ota 
0 —= (hg 1) ses 0 0 0 a3 
0 0 Oh cas —An-2 1 @) Gos 
0 0 0 oy 0 — Oy -1 1 | 
0 0; 0 0 0 Shs ts 
aie 21.0 0 Ov e.0ce ont 
a GM —l O On Of 0 


An-1 0 0 0 ae 0 9 ST | ze 1 


Gps re tt =O SD We a a 

Cire eee hws es Ol). 0.) 
Goee Get — 1] Oo ost 0-0 
a3 0 As 5 1 0 0 

An—1 0 0 0 OR =i | 

Dn Oy Gert UO Oa Oe | 
Yn Cero le On,» O Gear 
0 a —1 0 40 
es 20) 


Da 
150. The numerator and denominator of - can be trans- 


formed into continuants, and thus the fraction F’ can be 
transformed into a descending continued fraction, as follows: 


186 THEORY OF DETERMINANTS. 


Multiply the last row of p, by a,_;, and subtract from it the 


(n—1)th row multiplied by a,; then multiply the (n—1)th 
row by a,_2, and subtract from it the (n—2)th row multiplied 


by a,,; andsoon. Then 


Pepe ty it ee 0 0 0 
0 dga;+ag —ay O s+ 0 0 0 
0 —Ayag Agdg+az —Ag**s 0 0 0 
0 0 0 QO = —An—24n-1 An—19n—-2F%n—-1 —%n—2 
0 0 0 0 + 0 — Ay 1%, AnOn—1 + On 
Para= : 
On—-14%y-20n-g °°* Agag ay 
Similarly, 
Ay ~] 6) 0 wee 0 0 0 
— (A Cp0 + O9 ay 0 eee 0 0 0 
0 — AoA Agay-+ ag <Oo e's 0 0 0 
0 0 0 heey: —An—-2 An—-19~2 te 
On—-1 + ays; ; ed 
0 0 0 0k ae Onn 
n-18n, +a,, 
Pn == “ 
On-1AOn-9Oyn-3 *** AgQy 
Whence, by 144, 
Pn 
Qn 
ide A} Ag M2014 : An —20n 3%n 1 Ayn —1Oy 90, 
Cy, —ApQy+ag —Agzt,+a3 —On_10,-9-+O,_) On Oy, 


the descending fraction sought. 


APPLICATIONS AND SPECIAL FORMS. 187 


Alternants. 


151. Consider the determinant 


as 2 cy 

Peery Leaky thy = tise oT 
je POR (a 
oath 4h Cie (i? 


and the product 


P = (dg—y) (4g—y) (g— A) + (Ap — Ay) 
X (g—Ay) (4—g) +++ (An—Ae) 
X (@y—Ag) +++ (Gn—Gs) 


x (Gy, Tal On—1) 


of the 5 (1) differences of the n different quantities involved 


in A. This product is called the difference product of the n 
quantities a, d:, +--+ @,, and for it the notation €3(a,, dy, ds, +++ dn) 
has been adopted. 

The reader will remember that the square of the difference 
product was denoted by €(a, de, «+: a), and thus the difference 
product itself is very appropriately designated by @ (a, da, «++ a, 

We shall now show that 


A=P=2(q, dy, +++, A,). (1) 


If in A we put a,;=a,, A vanishes; hence A is divisible by 
each factor of P, and hence by P. Again, A and P are each 


polynomials of degree 5 (n—1), and therefore 


A= AG (ah, Cg, Agy ***5 @,,)'5 


where A is a factor independent of a, a, +++ @,. From the 
special case 


188 THEORY OF DETERMINANTS. 


Az | 1 nen 
1 yy? | = (A2—) (3-4) (Ag — Ae) y 
Leis ae 


we see that A=1, and thus the truth of (1) is established.* 


152. A of the preceding article is evidently an alternating 
function; for the interchange of a, and a, amounts to an 
interchange of two rows in the determinant, and hence changes 
its sign. A is accordingly an Alternant. In general, an alter- 
nant is a determinant in which each element of the first row is 
a function of #,, the corresponding elements of the second row 
the same funetions of «,, and soon. Thus 


fit) fol) «+ f(x) | =ALAC), Se) + Fakta 
Ai(@2)  fa(%2) +++ fu (%2) 


A= 


i 


ae J2(%n) Nie Jala) 


is an alternant. 


153. We can easily show that 


A=/1 Fi (2%) So(%) ches Fete = AG (a, Hey oe i 
1 fi(@2) — fa(®2) +++ Sn-1(2) 


1 Ai (&,) So(&n) r phe Sion) 


where f.(z) is a function of the rth degree in x, and X is the 
product of the coefficients of the terms of highest degree in 
the several functions. For subtracting the first column mul- 
tiplied by the proper number from the second, we reduce the 
elements of the second column to 1%, 1%, 91%, +++ 12,. 
Then subtracting the sum of the first and second columns, each 
multiplied by the proper number, from the third column, the 
elements of this column become p,.2,", 2%", +++ DP .%,. Pro- 
ceeding in this way, we see that finally 


A = AZ (a, Xo5 mint eas 


* See also examples 6 and 7, page 37. 


APPLICATIONS AND SPECIAL FORMS. 189 
where A == 1° Pore? Dns 


For an example, putting 
x(a—1)(a—2) --- (a—r+1 
Bec) = Niza MG iad C3) 


we have 


S{1 fie) Alm) --- fan) (84,5 aay ++ ay) 
IAC) < Blm)e-- feledte Chie ia et 


| 1 Ji(®n) J2(&n) tar Fn—1(%n) 


154. Every alternant whose elements are rational integral 
functions of 2, 7, --- 2, is divisible by ((a, 22, a, «++ %,q), 
and the quotient is a symmetric function of the variables. 
For the alternant vanishes if 2,=a,, and hence is divisible by 
®;— %, and thus by ¢3(a,, a, --- v,). The quotient must be a 
symmetric function, for the interchange of a, and a, changes 
the sign of both dividend and divisor; therefore the sign of the 
quotient remains unchanged upon the interchange of two of 
the variables, and is accordingly a symmetric function. We 
shall now actually perform the division just considered. Alter- 
nants whose functions are powers of the variables are called 
simple alternants, and are of frequent occurrence. We proceed 
first to the discussion of simple alternants. 


155. The quotient 


r. sue “7 ee! 0 . Dy) n-2 
}1 a 7 AGS Lay Ssh =** gt Vee) 


pd of Oh gr eas widiees ) ES 
1 Wg, °° eye Xo" | $ (2, 2s 4) is O(a, Woy *°* Wy) 


n—2 


may be developed as follows : 


Expand the dividend A in terms of the elements of the last 
column, and we obtain 


190 THEORY OF DETERMINANTS. 
Aa’, (oe 2X", 08 La a5 Xn ¢) 


UNE) 
ae ae dx,2 dz,! 


Now, it is evident that each of the minors in this expansion 


is a difference product. 


Thus 
0 2 —2 
Le sty Seo Sees oe 


dA 
= athe 1 n+r 
des (1) 


> 2 pn—2 
l °4,5) Ue 


2 an—2 
1 a Vr) gee, Vr 


120g 1 eee 
=(—1)"" (a, Voy +2 Uy Uppy °°° Tq)» (2) 


Substituting in (1) the values of the minors as found from 
(2), and dividing both members of (1) by €3(a%, 2, +++ %), we 
have a series of terms, of which 

(—1)"*"x,! 
(%,—2,) (@,_1—2,.) eh (%,41—2,) (%,—2,_1) Pies (X,— 2p) (%,—2}) 


is the type. Thus we find 


a ve 2 sae) = 
ryan (— Lyr xf 
~1 (%_,—2%,) (Bn_1— By) 2° (p41 —2,) (B,p— 2,4) 2+ (— a4) 


x,! 


or —— ee eee 
(%,—2%,) (%,—2,,_1) es (X— 22) 


Vo! 
(X,_—2Xp) (X_g— Bn _1) eee (X_— 23) (X— 2%) 


All 
Xn—2 


(Xp_2— Bp) (X,-2— Up, 1) (2X, g— 2X, 3) bes (X,-2—%)) 


a 


Terps st 


APPLICATIONS AND SPECIAL FORMS. 191 


q 
Ln 1 


(%,-1—2n) (&,1—%n_2) (X,_1—®,_3) Sook (®p_-1— 1) 


4 


Ly! 
(Xp—2n_1) (X_—Wn_2) tans (X,— Xp) (&,, — 2) 


of 


For an illustration, we have 


Ve2e AG) pls 204 
Ras ek Lage os 
1 5 625 Ve 20 
16 81 625 
ahs 


ee ee 5 = 69 
(@—5)@—3) | @—5)G—2) (6-8) G—2) 
156. With the help of the preceding article we may reduce 
the quotient 
A (ay, Wy 3" ay ini Lr! 
Ct (yy Vqy **+ Ln) 


to the sum of two similar and simpler quotients, as follows: 
Since 


0 1 2 — 0 1 2 —2 —l 
A(x; 9 Ue 5 Uy 5 °° ets Xn*) oad CAL, 3 Ug 4 Xzy °°° Ln—19 Lye ) 


Sta Bee ae OY (a —m,) I 
Lig Pilg ge Og tg) (ty, 
(1) 
1 Uy 1 | a ee nt ar (py — tb, 
LS eee har eee 0 


we have, after dividing both members of (1) by €'(%, ®2, +++ % 
in accordance with 155, and striking out the factor common 


to numerator 4nd denominator of each term in the second 
member, 
— 0 1 2 —2 .q—l 
A(x’, fed ae a* S's ate 2,1) oe A(X; 5 Uy U4 °** Dengan ) 
C(x, Xo5 Lares Ln) G( ay, 0 aie Xp 


192 THEORY OF DETERMINANTS. 
La Cees 
== Ht Ch eh _ROVOROQDNVYDamaSS 
(%— 2%») (%,—2g) +** (%—%n—1) (X_— oat res (2— 23) (%—%) 
ae 
a ooo 
(%,-1—Wn_2) (2%,-1—%n_3) de (2,1 —%) 


But the sum in the second member of this equation is, 
by 155, 
a fe yy -1 s 
Hy tee 8 TE | CEO, Og, 298s Dara 


2 pee = 
Xo Lo” ALI 2 3 Xo" 1 


p nd an—3 aq—L 
1 Ly-1 Un-1 *** Xn-1 Xn—1 


Transposing, we have 


A (ay, Wy'y Wy", +++ Wndy Wn!) — aA (Hy", Wy'y Wy’, 22° Vari, UE 
O(a, Xo, 29° Uy (a, Uo, *** Ln) 


a 0 yt a 2 ai —3 g—]J 
A (a, 9 Uo Uys °° Lyd, Vy 
j 4 
C (x, Xoy °°° Da) 


b) 


which is the desired reduction. For example, 


1 aera Leche aa 
Loy oe Mets: Fos. 1 
Leta Le Ce Litay; 
rN = Se. 
G@, 2) "C@y) Oey 19 
1 aor lr 
le Re a 1 
1 ae LOS eae oe fae y : : 2 
Ay, ee ee AT NN == RD 
G(X, Y, 2) Cha, Y, 2) C(x, y) ao a 
The student may show 
Wea " 
ee ne 
‘A 5 
fare SY 8 Sa’y + xyz. 


O(a, y, 2) 


APPLICATIONS AND SPECIAL FORMS. 1938 


1 a2 2 
Dvag- 9 
Ter oe 2 


————— = Sart + Sa®y + Say? + Sayz. 
O(a, 2) : 

157. Since every term of (a, a, --- 2,) contains a permu- 
tation of all the powers of the variables from 1 to n —1, each 


term is of the oe 1)th degree. Similarly, every term of 
A (a), %', #3, -+-, on-?, wf) is of degree hee bee ee ae +q. 
Hence every term of 
Q= A (2,", Xq'y Ley **%5 Uety Uf 
(a, Voy ty Vn) 


is of the (g—n-+1)th degree, as is illustrated in the exam- 
ples of the preceding article. We shall now show that every 
possible term of the (¢—n-+1)th degree in the variables is 
found in Q, and that every such term is positive. That is to 
say, the quotient Q is the complete symmetric function of degree 
(q—n+1) of &, %o, +++) Xp. 

Such a term of Q is 


— »38m0 m6 7 
eee wy Ly Ws 110 D9 Bin_1 Une 


By successively applying 156, we develop Q so that the 
terms containing @,, %, 2,1, %%,%,1%,-2, etc., are at once 
distinguished. In the first place, 


Ae al Mb an —38 —1 0 i 2 —3 —2 
Q ae A(x; 9 Wy 4 Uy y tr%y Uno Vay egies » Ly 4 Uy'y *e+y Und ni) 
=a ao 4 oe e . Qe 
“(x, Xo, ***%5 Xn—1) Ct ( a, sr | Ln-1) 
] 2 —8 —3 - r0 1 2 —8 —1 
eA(a,, Vo ,%3, 00°, Un 9s wes) xf "A(x; »U9,Xsg °° *,Un—2 Uni ‘ 


- spevep 


G(X, oy +25 Vn_y) C3 (ay, Voy **4 Ly_1) 
a est} 
Ul 
The second term, 
7, A(a?, wd, cg, r+, at 8, at?) Q 
meres. Ys Tei ae Sean 


“3 (2, Xoy *%%5 os) 


194 THEORY OF DETERMINANTS. 


contains the first power of x,; hence we must look for 7’ in 
Q@,. Applying 156 to @,, as before, we have 


0248 a es ee By ae 
Oi=o becca EP el ed Vn A(X’, Hy, Ug ***, Uns, Ag 


ile 
O(a, Vay **%5 ois) 4 (2, Xey **%5 Xn—2) 


: 2 Gee ene Bie eS 
are aa | (xy » Ue 4 Lz y 22%y Un 35 UE =) 
O(a, Voy ***5 Xn—2) 

etl A (09, ad, we, -+*, ie 
1 19 %25 U3y 9 Un -2 “ 
ip el et an rs es + a7 |. 


G(x, Xo, **%5 axe) 


In this expansion we must look for 7’ in the third term 


ae on WC es We Scum? 2 pate nie a Q 
2. wae ae — Qe 
(4 (Xj, Hoy ***y Xn—2) 


@, may be expanded as before; continuing in this way, we 
finally obtain the term 


. of A (x; x3") a): 
O(a, a2) She 


for the coefficient of 2,,x7,_,27, »--- 2f contains only a and a», 
and is of the third degree. Upon performing the division, and 
multiplying, one of the terms is 7. Since 7 is any term, the 
proposition is established. 

Employing the notation H, for the complete symmetric 
function of\the rth degree, we may write the result of the 
present article 


7 
1A oes Poppet eke 


Ae ore ie. ome eee 
TR = Hina iy Be 5) 
i EP) A en . 


or simply eas 


For illustrations the student may refer to the examples in 
the preceding article. 


* With this notation, H)=1, H_-=0. 


APPLICATIONS AND SPECIAL FORMS. 195 


Again, 
Peewee oF 
SU ia 
i wy fa ee a 
i oh et lan 


——_____— = Hf, = Sx* 4+- Say + Say? ; 
C(x, y, 2, t) et a oe eae 


158. From the two preceding articles we have at once 


FT, (%, Vos **"5 Xn) a €,LL,. 1 (2X), Voy **"%5 Ln) ae HT, (2&1) Hoy ***5 Xy1) ° 


(1) 


Whence we readily obtain 
HA, 1( Hj, U5 ° °°, Bn 41) = Xn 41,9 (Hr, Le, °°) Un 41) +H, _3 (ay, Lo, ***, Vn); 
2» H,_1( 2), Xo, -+*, X,) = H1,_ 1%), Xg °**, %n41)—Xn41H,-o(%, Ho, ***, Uns)» 
Substituting in (1), 
HT, (4, Xyq +*+y Uy) = %,[H,_1 (1) Wey °**5 Vir) 
Hitt, _o( 15 Ley ***5 Lary) | +H (ay5 Woy **5 La)» (2) 
Similarly, 


HT, (%, Xo5 °%"5 Bair) =e CP pp w aN Ce Xo5 ***5 Cot) 
>) %, H,_9(%, Voy **%5 Bn+1) | + Hf,(%, Vo, °° "5 ose) (3) 


From (2) and (38), 
FT, (15 X25 °° Xn) — HH, %, °° Xn —1n +1) 
ai (x, — 2 doen go Bee (2, Wo, **° Xn+1) e (4) 


159. If any alternant whose elements are powers (simple 
alternant) be divided by the difference product of its variables, 
the result is expressible as a determinant whose elements are 
complete symmetric functions of the variables. That is to say, 


A(a;, Wy | shee) cis ee Hg vee Fy 
¢? (21, Hog 29° Ly, 7 Hg_1 a ane y_1 


ja frees | Ag_n+1 aes y—n+1 


196 THEORY OF DETERMINANTS. 


This may be proved as follows. For brevity we employ 
determinants of the third order, but the method applies, of 
course, to determinants of any order.* In the alternant 


A(ai, ab, 53) =| at mm R 


he el ce 
subtract the first row from each of the other two, then remove 
the factors (#— 2%), (#,—2#,). Afterward subtract the second 
row from the third, and remove the factor «;— 2, employing 
equation (4), 158. The result is 


A(x; , att ay) FT,,(%;) Hg (2,) HI, (&;) pa 
OCLs Ces Ue) FIg-1 (15%) — Hpa(%,%2) — -Aya( a, 2) 
H,2(%, %,, %3) LHg—2( 2%, Xe, ag) Ffy_2(X, NXg, 2g) 


The determinant on the right we now transform as follows. 
Add the second row, multiplied by 2, to the first, employing 
equation (1), 158, and obtain the determinant 


LT, (2X). Xo) 1p (x, Xo) FT, (2%, Xz) 
H,1(%, ©2) Fp_1(a%, Wp) Hy _4( 2, Wp) 
H,-9(%, X95 3) ET g-9(2y, X25 3) HT,_9(%, Xo5 2%) 


Now add the third row, multiplied by a, to the second, 
again employing (1) of 158; finally, add the second row, 
raultiplied by ws, to the first. 

We then obtain 


A(ay, ats , 3 ) ry H,( 2%, Uo, a3) Ag (21, Vos a3) HI, (&, Voy Ws) , 
E(@yy ®, 3) 1,1 (%} 2X3) Hg_1(%), Xo, %g) Hy1 (a, ao, Xs) 
F,_9(% 25 Uy) Hgo( 2), Le, 3) FHy_2(%, Xe, 2s) 


as was to be shown. For an example, 


* The mode of proof here given is due to Mr. O. H. Mitchell, American 
Journal of Mathematics, Vol. IV., page 344. 


APPLICATIONS AND SPECIAL FORMS. 197 


Peat = upe enn a oe 
dD bY BF 1 38 of 
ola Live ome 
= abeli(a, b,c) | Hy(a,b,c) H;(a,b,c) H,(a, b,c) 
H_, 2 HI, 
f_, fH, Hf, 
=abcli(a,b,c)| H, HA, 
Hi, fi, 
= abe(a,b,c)| Sa? + 3ab Sa*® + Sa*b + Sade 
za ya? + Sab 
=abel(a,b,c)| —3ab — a*b—2dabe 
Ya ya? + Sab 
= abcel(a,b,c)| — Sab abe 
sa = — ab 


= abel? (a, b,c) (Xa*b? + Sa’be). 


160. Form the product 


Peel Om Oy | XL oP? af ee wm 1, 


Qo, Ogg °°° on a ie 


Be Oe aes 


Ani Ane aia Onn 


changing the columns of the first determinant into rows before 
multiplying. If we put 


Ie) = hy, 0”) hy, UP? eee Ly ay ® HE Anys 
we find ey) 
P=(-1/ | Han | (a4, Boy ***y Ly) 


=(| fil) (m1) + Sr(%) |- 
Fil%e) faa) e+ fn (Ha) 


Si(@n) Fa (An) le Fkte) 


198 THEORY OF DETERMINANTS. 


If now 


SF, (&s) = (x, ans Yr) ie 


we must have 


la, |= 


mee (ce ee ; (=) @ele>, Rea fle 
1 CS ea9 C5") (Cena em 


where 


CO) See () — 2) =) (pe 
Che q! 


But this last determinant evidently equals 


EK (—1)*? B (yas Yor Yor °°") Ya)» 


where # is the product of all the binomial coefficients of order 
n—1. We have, accordingly, 


(%—y)"* (B® — Yo)" (1 — Yn)” * 
(w_ — ar (@— Yo)" vee (BQ — ger 


(Wp mae he (Ln res Yo)? ~~ (a, a aa “oa 


— KG (mx, Vo, Ug, ***, Ln) Oy, Yoo 39°" Yn) - 


If now 2, =y,, we have £(a, 2%, W3, ++: 


,v,) in the form of a 
determinant. 


161. Suppose now that aj, ay, :: 


-,a, are the roots of an 
equation 


Fe): =0; (1). 


: Cs pen » (3 em (“5 my ve (= . 


APPLICATIONS AND SPECIAL FORMS. 199 


Then €3(a,, a2, a3, +++,a,) is the product of the differences of 
the roots of (1). Square this determinant, obtaining 


€ (aj, a, ses a Cy, ) = 1+1 ieee +1, ay +a, +--+ +a, 
a) + dy wm eoiache el! aptaz+--»-+a,? 
ay + ay eee Hp ay ajay vee + Oy 


-1 =t = 
Oy agh eee tae) afttast -. +a," « 


oa -. ast 4. fox Z + az! 1 
ay. ag 4 = a,” 


age + at! ae: ate +a,7+1 


ap” tas? Se +4,2" 


soe One eres gn enden es K . 
pI Sa eS Og 8 Ok ie 
Ca Se Og SE Sy 


Sn—1 Sn Sn4t cee Son 


where, as usual, 
S, =aj +azy+---+a,. 


162. The preceding article gives us an expression for the 
square of the differences of the roots in terms of s,; We can 
also readily obtain an expression for the sum of the squares 
of the differences in terms of s,; as follows. 


We have 
te ees 12 sl) 8, foo 2 
Co ey w= 8; 8, Gt we 
by 58. 


163. We shall conclude our discussion of alternants with a 
theorem on the reduction of alternating functions to alternants.* 


* “ Reduction of Alternating Functions to Alternants,” Wm. Woolsey 
Johnson, American Journal of Mathematics, Vol. VII., page 345. 


200 THEORY OF DETERMINANTS. 


Any function of the form 


di(a, bed +++ 1) do(a, Bed +++ 1) +++ hyn (a, bed «++ 1) 
di(b, acd +++1) — py(b, aed +++ 1) +++ of, (0, acd «+. 1) (1) 


di(l, abe +++ k)  do(l, abe +++ k) +++ h,(l, abe +: k) 
is evidently an alternating function of a,b, c¢,---1, if 
h(a, bed «+ 1) 


denotes a function of the n quantities a,b,c,-+--1, which is 
symmetrical with respect to all the quantities except a. If 
each element of this determinant contains only the leading 
letter, (1) becomes 


fila) fea) fa(a) - fn() |, 
SOY IO) Ie) eee 


‘ = (2) 
AQ AO AO o> LO 


an alternant, which we represent, as usual, by its principal 
term, 
[AC@), A); BC); vee f, (0) |. (3) 


Now, if the principal term of (1) can be separated into parts 
of the form (3), then the given alternating function (1) is 
equal to the sum of the alternants represented by these partial 
terms. This is proved as follows. Since an interchange of 
two rows of (1) is equivalent to an interchange of the corre- 
sponding letters, any term of (1) can be obtained from the 
principal term by a suitable transposition of the letters, and, 
similarly, the corresponding term in each of the alternants may 
be derived from its principal term by the same transposition 
of the letters; hence every term in the expansion of (1) is 
equal to the sum of the corresponding terms in the expansion 
of the alternants. 


APPLICATIONS AND SPECIAL FORMS. 201 


Accordingly, if a determinant of the form (1) is expressed, 
as usual, by writing its principal term in (), with commas 
between the elements, we may erase the commas, and treat the 
expression within the () as an ordinary algebraic quantity. 

Thus, 


bed 1 a a?|= A(bed,1,¢,d?)=A(a°,d, 2, d) = (a,b,c, da). 
ol DS og 
wapeet 6. Cc 
fool .d d* 
Again, 


1 ¢+a’*? b+ ca 


ino +c Pa 
1 a+b c¢+ab 


= A(a°, 0°, ct) +A(a’, d°, c?) + A(a, b, c?) +A(a5, b, ©) 


=— A(a’, b,c) =—(a+b+c) G(a, b,c). 


Functional Determinants. 


164. Consider the following n functions of the n independent 
variables 2, X, +++ X.- 


Yi = Fi{ ty Woy °**s Xn) 
faieed ae (1) 


Yn = fi, (Ms Loy ***5 Ln) 


These functions will be independent if for every set of values 
Of 4, Yo": Yn equations (1) determine one or more sets of 
values of 2, %, ++» 2, so that these latter variables can in their 
turn be considered as functions of the n independent variables 


Yr, Yar *** Yns 


202 THEORY OF DETERMINANTS. 
Differentiating equations (1), we have 


5 of, 8 
dy, = hae, + ae, 5 “<a 


n 


dy, = sHae, + Pao, +. ay an a @) 


of, of, ofr 
ly, = "da =~" dx, ooo "da, 
ay ae ee Sa, ‘Yar 5 


nr 


Regarding equations (2) as a system of equations for deter- 
mining dx, da, --- dx,, the determinant of this system 


oy on oe on = 8 (%5 Yor **"5 Yn) = J 
0%, day dx, 0 (Wy, Way +*+y Ln) 

a Ola ary a Ou 

0%, ON%y 02, 


Bp Ode a coe 


is called the Jacobian of the given functions y,, Yoy +++ Yn» 
Or, in other words, the Jacobian of a set of n functions, each 
of n variables, is the determinant | %,,|, in which the element 
k,, is the first derivative of the pth function with respect to the 
gth variable. Thus, given 


yy =a/+2bet+c , y= aye’+2bdet +e. 
The Jacobian 


8 (%; & (Yi Yo) 50) 


az +bt bz +ct Aly, bz a 
38'(z,*t) 


Az+ot be+et be z Yo be+et 


a ad C) 
mh bh Cy 


dil Oe aa) geal geet 
8 a be hd tct ae 


'Y be+tet gq 


APPLICATIONS AND SPECIAL FORMS. 202 


165. If the functions y,, y, --- y, are not independent, but 
are connected by a relation 


P (Yrs Yoo ** Yn) = 95 (3) 


the Jacobian vanishes. 
From (3) we have, by differentiating, 


Bb Ay | 8b Be gg Ob By 


0 
dy, 3%, dy, day by, Sa, 
dd 8% , 5h . d% op by 
—_ —_-_ ss eee See ae O 
bY, de a SY, SMe pear SY, Oxy i (4) 


dp | oth 4 8h | dy op | dy 
Sad ne eben ow fonliesl eee — . v* — 0) 
OY dx, iy dy 2 52, = 43 SY n 82, 

From their mode of formation, equations (4) are simulta- 
neous. Hence the determinant of the system vanishes by 77; 
ai igear( 

We shall show presently that if the Jacobian of a set of 
functions vanishes, the functions are not independent. 


166. The Jacobian of the implicit functions 


F, (2, Voy *** ny Yi5 Yos *** hi 0) ] 


Fy (15 Loy *** Lay Yay Yoo *** Yn) =O ( (5) 
Fe aN ah enenhy = 9. 
is found as follows. 
Equations (5) yield 
A 


~ ba, dx 9x, Syn dm, 8Yn OM 
(t, Kee 1, 2, ae mye 


9204 THEORY OF DETERMINANTS. 


Using equation (6), we find the product of 


SM 8R, | OF | and ay Bye Bh 
by, OYs Wis &Yn da Oa Bae 
Sy, 82 = Yn Sat, Ody, meters 
OF, oF, OFe] Bn Bt 8 
54; -8Y_ OY | 3%, Oy bX, 
to be 
(— 1) ebony OF; |. 
Sa, dt a 
of, OF, OF 
Sati Oka ao eh oan 
oF, aF, | a, 
da, S% Sap 
Whence 


_ 8(ty Yas Ya) ¢_ yy Fv Py: Fa), 8(F, Fe os a) 
J= = ( 1) a ee 
8( 24, Xo5 eee Ly) 8( ay, V5 eee D) 0(Y5 Yas eee Yn) 


(7) 
If in (7) we put n=1, we get 
_3F,_ Fs dy 
Sa, dy, da, 


a well-known formula. 


167. If in equations (5) we consider %, %, ++. %, as func- 
tions of Y, Yo, +++ Yn» We Obtain, as above, 


6( Fi, Py, +++ F,) _ OF, F, «++ F,,) x O(a, Vo, ++ Wn) (8) 


(21) 2a Be sey Es ee 
b(%15 Yos *** Yn) 6(a, Voy *°* Ly d(%, Yo, °° Yn) 


APPLICATIONS AND SPECIAL FORMS. 205 


From (7) and (8), 


d(%, ee ee Yn) (a, Woy 209 & ) 
eK = 1, 9 
6(%, Xo5 6 s'e Cad d(%1) Y25 eee Yn) ( ) 


168. Again, having given the n+ ) functions, 


BP, (%y Woy +++ ny Yay Yay oe Ynip) = 0 | 
F, (2, Hoy *** Uny Yrs Yoo °°° Ug) 0) t 


(10) 
Ftp (5 Hey °° Uy Yrs Yor v** pry, = 0 


The Jacobian 


4 fe d(%1, Yo cis Yn) 
6(2, Loy ++ Ly) 


of the first » of these functions is found as follows. Differen- 
tiating equations (10), we find 


bF, _ 8F, bn , BF, bys 


OF, OY nap 
coeth aan dees oe 


i : b 
nip) O&p () 


(==, 25° n+p; k= 1, 2,+.5 ny, 


Now multiply together 


A=| oF OF, MW lead ies oh Yo 8Yn | 
So ees CNG) td wees da, 5a, 9&2, 
OF, 48F, oF, Oy, OYe OYn 


Oy, dYo ae BU avy 8X, dxy ae 82 


n+p aes Ole by dYs , 8Yn 


OP ere Oey 8% 
Byse cect a 8 OY, San,2) Bee.) oe 


first writing J as a determinant of order n+ p, thus: 


206 THEORY OF DETERMINANTS. 


ba, 62, 
Bh 
bX, 82, 
i 8Yn 
bX_ OL, 
0 0 
0 0 


Calling the product P, we have 


P=(-1)"|8h ah 
a, 52, 
oF, bf, 


62% 62, 


yrs.) 


ceed 
02, ba, 


=(—1)” 
( 8d (2%, Voy *** 


since, by equation (6) for k<n, the element a, of P is — 


and for k>n, ee 


Yi 


We have, accordingly, 


J= 


~ 


jy n+1 éy n+p |° 
32 On; 
Ont 8Yn+p 
dA bX, 
éy n+1 oy n+p 
62, Ou, 
1 0 
0 1, 
OF, OF, 
dy n+l . dy n+p 
oF, oF, 
oO] n+l dy n+p 
OF. n+p OF, n+p 
] n+l by n+p 
Ftp) 


0(F, Pa, + 


} 
vay Yntis Yn4+29 Bar Yn+p) 


169. Suppose equations (5) yield upon solution 


y= 1 (2, Ly, 24+4 Vy). 


oF; 


——— 


vy 


(c) 


° 
’ 


APPLICATIONS AND SPECIAL FORMS. * 207 


Solve (c) for 2, and substitute this value of a in the 
remaining n—1 equations; then Ys, 73, «++ ¥, become functions 
Of Yi, %, +++ %,. Thus 


Ys = b2(Y15 Voy *** Xn) : (d) 


Solve (d) for 2, substitute the result in the remaining n — 2 
equations ; then ¥3, %, --: ¥, become functions of 


Yrs Yay Vg °8%y Une 
Thus Pe 3 (Ys Yay Uzy **"%5 Xn) . (e) 
Solve (e) for x3, substitute as before; and so on. 


We obtain the equations 


9 ging (2, Voy °** Xn) = 0 
Yo — o(Yry Voy **° Ln) = 
Ys — P3(Yiy Yos Lay **° Xn) —) Pe x (11) 


Yn < bn (Yrs Yo ie Yn-13 ate Ln) at 0 J 


By 166, ics 8 (4%, Yas ‘= Yn) 
0( 2, Wo, °*° Yn) 


= (-1) 
_ ad _ 8b) _ 3dr. _ 8th 
Say. Say a Sa, 1 0 0 0 
0 do __ 8h os __ doh2 de 1 0 0 
dx, O28, dx, oy; 
R dhs 3h) | __8¢s _8¢3 ; 
025 62, by, OY 
3b,| | _ 8b, _ by 8b»... 4 
‘ : : d@,, oy) Oye OYs 


Sgr , Sho , 83 ,,, On, 


208 THEORY OF DETERMINANTS. 


That is to say, the Jacobian of a set of functions Yy, Yo5 *** Yns 
each of n independent variables 2, Xa, +++ %,, ts expressible as a 
product of n differential coefficients of the functions 4, $2, +++ dn, 
where p, is a function Of Yy, Yor *** Yp-yy Bpy 22% ye 


170. The result just obtained may be employed to show that 
if the Jacobian of a set of functions vanishes, the functions are 
not independent. 


For, if yah Bb Obs 


5d; 
LPO) 
ba; j 
where 7 has one of the values 1, 2,---n. But if Ode =a Oe 
does not contain 4, 7.e., Us 


Ye = Pi (Yas Yoo 2° Yi-ry Visa, *** Uy) 
Also Yisr = Dizi Yrs Yoo *** Yis Vig, *** Uy) 
From these two equations, 

Yirr = Visa Yrs Yor +++ Yin Citas Vinay + Ln) 3 


therefore y;,; does not contain %,,;. In the same way we may 
show that ¥42 does not contain 2%,,, and so on. Hence, 
finally, 

Yn = Wn (Yas Yor *** Yar) 3 


or ¥, 1s expressible as a function of the remaining n —1 fune- 
tions, and hence the given functions are not independent. 
For example, if the given functions are 


(1) wsat+y, (2) v=ae—2, (8) w=aytozr—yz—2, 


J=/1 1 yt+z 
oa L—z 
0 —1l w—y—2z 


APPLICATIONS AND SPECIAL FORMS. 209 


J evidently vanishes. Accordingly, (1), (2), (8) are not 
independent. That the given functions are not independent is 
easily shown directly as follows. We readily obtain 


yY=uUu—v—z% ..w=v(u—v), 
as was to be shown. 


171. If the functions y, yo, +++ y, are the » partial derivatives 


ef sie Ase ony of a function I (hy Uo, o*° e's the Jacobian 
Ol, $s 82, 
Eg 8 OBS SE a 
On, - 62,02, 620%, 
PS fee vy hp ened it 
OMaod oe Ole §252,, 


Pee Sa eae STE eh al 


62,00,  O0L,0%e ox,? 


is called the Hessian of (a, %,++-%,). The Hessian is a 
symmetrical determinant, since 


of - oy 
62,02, 02,00; 
Shea ht OF 


If the derivatives =, 
0%, 0%, | (0%, 


tion, with constant coefficients 
of 


Jina ans doy ee cies 0, 
OW, 62, 


, are connected by an equa- 


6 

Cy eh + > 
62 

the Hessian must vanish. 


172. Let fi, fi, «+f, be » given functions of the same 
variable x. Suppose the functions are connected by the linear 


relation ; 
hf, + defo + A3,f3 + et Ontn = 9, (1) 


210 THEORY OF DETERMINANTS. 


in which a, d:,+:: a, are not functions of x. Differentiating 
(1) successively n —1 times, we have 


afi +f) +a;,f) +: af, = 0 
Hee ape seca ah =0 (2) 


aft to auf Po anf * Uy fat = 0 


Eliminating a, a.,--- a, from (1) and (2), we find 


Si Sr Ss sar In = D( fis Sos Sas + f,) = 0. 
3 ! by eee his 
jn jr ip ee aL (3) 


1 eo oe Ssh los f ge 


The determinant of (3) has been called the Wronskian of 
Ji Js, «+ Sue We see from (3) that if the functions f, fi, --- Jf, 


are connected by a linear equation of the form (1), the Wron- 
skian vanishes. 


173. If we denote the given functions by y, y2, +++ y,, and 


the derivatives by 4), Ya, +++ (%-e., the second subscript denoting 
the derivatives), we may write (3) 


Yat Yn | = D (Yay Yas Yas *** Yn) =O 
Yu Yau anti re Vat 
Yi wath U8 ned Una 


Now y being any function of a, we find 


Y"D(Yry Yas Yas 8° Yn) =| MY (MY) oo (HY nal 


a wae Z (124) na 5) 


Yn cay ei (YnY n—1 


APPLICATIONS AND SPECIAL FORMS. Zit 


in which the subscript & of (yy), means the kth derivative of 
(yy). That is to say, the Wronskian of wy, yoy, +++ Y,y is 
the product on the left in (5). This is made evident by notic- 
ing that since 


(YY = Yay + yy", (YiY oe = YoY t+ 2yay' + yy", 


etc., where y', 7'’, ---, are the successive derivatives of y, the 
determinant on the right becomes a sum of determinants, of 
which the first is the product on the left, and all the rest 
vanish. 


174. We find 
AD (Yar Yor ***1 Yn) | 9 Yu tt Yana Yin] 
da i oS : ee. & as 


for in the sum of determinants which make up the derivative 
sought, all vanish except the one expressed in equation (A). 


175. If in 173 we put y= ne the Wronskian on the right in 
(5) reduces to A 


©. ~ Q.l-AOO-O] 
© @ 


@ = @ 
Yi/2 Yi/n-1 


6 
Yi/1 
Now 
y\ _ D (sy 2) (#2) D (Yr Ys), ) _ DY» Mn), 
i ic nN Yi/1 1 Yy/ 7 Yi 


ll 


aL THEORY OF DETERMINANTS. 
Then if we put 


D (415 Yo) = %y D (Yay Ys) = % +++ D (Yay Yn) = ns 
we get 

1 
D (Yay Yas **° Yn) yy? D (22, gy °** Z_)- (6) 
Sh! 

176. We shall employ the result just obtained to show that 
if the Wronskian of 4, %,+:: ¥, Vanishes, the functions are 
connected by a linear equation having constant coefficients. 
Suppose that 7, does not vanish, and since by hypothesis 


L (A: iano Yi) — 0, 


by (6) of the last article we must also have 
1 
(ey, R35 eee Zn) ones 0. 
Y1 
Therefore, by 172, the n—1 functions %, 2s, «++ z, are connected 
by a linear relation, 7.e., 


C9 % + 323 + ary hs Aye = 0. (7) 


Dividing (7) by y,’, and restoring the values of 2g, 2g, +++ z 


n? 


Yo Ys n jou 
“2 (2) + ats (2) + eat Be (=) = (8) 


Integrating (8), we find 
Ly Yy + Mg Yo + Ag Yg + +++ + On Yn = 0. (9) 


Therefore assuming that if the Wronskian of n—1 fune- 
tions vanishes, the functions are connected by a linear relation, 
we have shown that when the Wronskian of n functions van- 
ishes, the functions are connected by a linear relation. But 
the assumption is obviously true for two functions, hence the 
theorem is true universally. 


APPLICATIONS AND SPECIAL FORMS. val 3 


Linear Substitution. 
177. If the n functions (one or more) 


Sy = My B+ yy Wy vee H Aint, | 
So = Ag ®1 + Agy Xp +++ + Con Ly, | 
eee eee eee eee eee f (1) 
jf = Ay XY ate Ong + 9 =f Ann Uy, | 


are transformed into functions of %, y, --- y, by the following 
linear substitutions,* 


a Ele bn yt Dip Yeo oy get Din Yn 
eee ne ht (2) 

LC, = bin ae bn2Yo a ie % —- DanUn | 
the determinant |0,,| of the system (2) is called the modulus 
of transformation. If the modulus is unity, the substitution 


is unimodular. If 2, a%,---#, are independent, the modulus 
cannot vanish. 


178. If the functions (1) are transformed by means of (2) 
into 
Si = My YH My Yo es FM Yn ) 
ie = Moy Yi HF Meg Yo +> + Man Yn tm | (3) 


oe Fei Mny yy at Mne Yo +: ~+ Mann Yn 
the determinant of the system (3), 
|My, Mog *** Man | 
* The learner can understand the importance of linear substitution by 


noticing that such a substitution is the process involved in transformation 
of codrdinates in Geometry. 


214 THEORY OF DETERMINANTS. 


equals the product of the determinant of the given system (1) 
by the modulus of transformation. That is to say, 


| Min | = 1 an] X | Ow]. 
This is proved as follows. The coefficient of y,, 
Mey = Ui Oy + ig Dox + ++ + Gin Onxs 


is found by multiplying equations (2) by dq, dg, +++ Ging reSpec- 
tively, and adding by columns. Whence, by 53, we see that 


| My My © My|=1Ay Ay ++ An|X| Oy Oy = Oy|- 
Mo, Mog °° Men Gg, Ogg *** Coy Do Dan #+* Don 
Mn Mag *** Man Gni Ang °** Onn by Dig ae ie 


179. If f(a, %,-++%,) is to be so transformed by the sub- 


stitution 
% = Butit Bi Yo +--+: + Bin Yn | 
Xo = Ba Yi + Boo Yo toe + re Yn (a) 


Ly = Bruit a Pn2Y2 + a rae! 


.that Ye YE Ht HYy = UY Wy ves + yy 


the linear substitution is called orthogonal. The coefficients of 
an orthogonal substitution must satisfy the following condi- 
tions. 

A. Since 


Wyse te HY? 
= (Buti + Bis Yo + +++ + BinYn)? + (Bor 91 + Bos Yo + +++ + Bon Yn)? 
see eee eee et (Bath ae Bas Ya 
= (Pr + Ba +++ Bar)yt + (Bis) + Bos +--+ + Bre) ys? 
» + 2% 42 (Bu Bie + Bar Bo + +++ + Bui Bue) + +5 


APPLICATIONS AND SPECIAL FORMS. PARE 


we must have 
pes +B +e++68,7? =1 
Bui Bix + Boi Bo + coe BarPus = 0 (4, k= Be 25 “ee n). 


B. If we wish to return to the original function from the 
transformed function, we must put 


II. Yi = Bruit + Bo %e + ++» + Br Mr 
For from (a) we readily find 
Bri %1 + Boi %2 + +++ + Bri®n 
= (Bu Bu + Ba Bs + +++ Bri Bri) + Y2(Pr2 Burt Bo Bat +++ +B re Bx) 


+s + Yn (Bin Pri + Bon Poi ae Peconn pat) 


Now, by I., the coefficient of y,= 1, and the other coefficients 
vanish. 


C. The square of the determinant of the system (a) 
{modulus of transformation) is unity. 


For 
Bu Piz ° . =i 
Ba ee e+ Bon 
Bui Bus fs Bas 
|D,,| is a symmetrical determinant by 108; since, by I., 
Pee 1), ot 1s 
dhe truth of ILI. is obvious. 
D. B, being the minor of B;, in | Bi,|, we find 
By = Bx! Binl- 
For multiplying the equations 
2 i a7 eg a ce = 0, 


Ht. — — Bin |? = | in ° 


Ps Bs mp aes) Pus Bu = = i 


Bn Bu ay ia Bn Bu = panes 0, 


Pit J 4 


216 THEORY OF DETERMINANTS. 


\ 


in order by By, By, --+ B,,, and adding, we have 


Buy aoe Bu (Bu Bate + Pin By) ote ererte Bix (Ba Bi SM pr Bin Bin) 


a Mtr + Br (BiBat 51$ + BrnPBin) + 


But all the coefficients, except the coefficient of 8;,, vanish ; 
hence 


IV. Bu = Bi | Bin | . 


EH. By the preceding condition IV., 


(Ba Bu + mes + Bin Bin) | Bin | — Bi Bu + ee + Bin Bun- 


The second member of this equation is | 6,,|, or 0, according 
as 7 and & are equal or unequal. 
Whence 


| Bx = Be - ahs —- Bia 
Baka aE Bie Bre f+. -f- Bin Bun os 


l 


Ff. The following relation holds between the minors of the 
modulus of the orthogonal substitution. 


| Rlaieas py spay ate 2 


Brtin| = 'Binl X |B By s+ Br,|- 


aa at ae - a 
Bees Bes Saree Bie Ba B,2 oo Bea 
For, by 61, . 
By By eed 


Bi, = | 1c ie, | Brass r+1 By 41 $22) oe Beaa n |* 
By Bry he By, B49 r+1 Brig r+2 - 28 Ban 


6% r+1 Bs r+2 ne ee n 


APPLICATIONS AND SPECIAL FORMS. 2Q17 
Now, by IV., 


By By dee By, =| Bin |" X Py Pre cae Bi» . 
By By aos By, Boy Bo» a Bo» 


ep. .:. B.. Boe Reais Bs 


Whence, equating the second members of these two equa- 
tions, the relation VI. follows. 


MATHEMATICS. 89 


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UNIVERSITY OF ILLINOIS-URBANA 


512.9432H19E C004 
AN ELEMENTARY TREATISE ON THE THEORY OF 


PTTL 


3 0112 01708 


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